(i) Aox_(R)K~=K[X_(1),(1)/(X_(1)),dots,X_(n),(1)/(X_(n))]A \otimes_{R} K \cong K\left[X_{1}, \frac{1}{X_{1}}, \ldots, X_{n}, \frac{1}{X_{n}}\right],
(ii) for each height-one prime ideal PP of R,Aox_(R)k(P)~=k(P)[X_(1),(1)/(X_(1)),dots:}R, A \otimes_{R} k(P) \cong k(P)\left[X_{1}, \frac{1}{X_{1}}, \ldots\right., {:X_(n),(1)/(X_(n))]\left.X_{n}, \frac{1}{X_{n}}\right]
Then AA is a locally Laurent polynomial algebra in nn variables over RR, i.e.,
and is of the form B[I^(-1)]B\left[I^{-1}\right], where BB is the symmetric algebra of a projective RR-module QQ of rank n,Qn, Q is a direct sum of finitely generated projective RR-modules of rank one, and II is an invertible ideal of BB.
5. EPIMORPHISM PROBLEM
The Epimorphism Problem for hypersurfaces asks the following fundamental question:
Question 4. Let kk be a field and f in B=k^([n])f \in B=k^{[n]} for some integer n >= 2n \geq 2. Suppose
B//(f)~=k^([n-1])B /(f) \cong k^{[n-1]}
Does this imply that B=k[f]^([n-1])B=k[f]^{[n-1]}, i.e., is ff a coordinate in BB ?
This problem is generally known as the Epimorphism Problem. It is an open problem and is regarded as one of the most challenging and celebrated problems in the area of affine algebraic geometry (see [38,69,75,77][38,69,75,77] for useful surveys).
The first major breakthrough on Question 4 was achieved during 1974-1975, independently, by Abhyankar-Moh and Suzuki [5,86]. They showed that Question 4 has an affirmative answer when kk is a field of characteristic zero and n=2n=2. Over a field of positive characteristic, explicit examples of nonrectifiable epimorphisms from k[X,Y]k[X, Y] to k[T]k[T] (referred to in Section 2) and hence explicit examples of nontrivial lines had already been demonstrated by Segre [83] in 1957 and Nagata [71] in 1971. However, over a field of characteristic zero, we have the following conjecture:
Abhyankar-Sathaye Conjecture. Let kk be a field of characteristic zero and f in B=k^([n])f \in B=k^{[n]} for some integer n >= 2n \geq 2. Suppose that B//(f)~=k^([n-1])B /(f) \cong k^{[n-1]}. Then B=k[f]^([n-1])B=k[f]^{[n-1]}.
In case n=3n=3, some special cases have been solved by Sathaye, Russell, and Wright [73,76,79,89]. In [79], Sathaye proved the conjecture for the linear planes, i.e., polynomials FF of the form aZ-ba Z-b, where a,b in k[X,Y]a, b \in k[X, Y]. This was further extended by Russell over fields of any characteristic. They proved that
Theorem 5.1. Let F in k[X,Y,Z]F \in k[X, Y, Z] be such that F=aZ-bF=a Z-b, where a(!=0),b in k[X,Y]a(\neq 0), b \in k[X, Y], and k[X,Y,Z]//(F)=k^([2])k[X, Y, Z] /(F)=k^{[2]}. Then there exist X_(0),Y_(0)in k[X,Y]X_{0}, Y_{0} \in k[X, Y] such that k[X,Y]=k[X_(0),Y_(0)]k[X, Y]=k\left[X_{0}, Y_{0}\right] with a in k[X_(0)]a \in k\left[X_{0}\right] and k[X,Y,Z]=k[X_(0),F]^([1])k[X, Y, Z]=k\left[X_{0}, F\right]^{[1]}.
When kk is an algebraically closed field of characteristic p >= 0p \geq 0, Wright [89] proved the conjecture for polynomials FF of the form aZ^(m)-ba Z^{m}-b with a,b in k[X,Y],m >= 2a, b \in k[X, Y], m \geq 2 and p∤mp \nmid m. Das and Dutta showed [28, THEOREM 4.5] that Wright's result extends to any field kk. They proved that
Theorem 5.2. Let kk be any field with ch k=p( >= 0)k=p(\geq 0) and F=aZ^(m)-b in k[X,Y,Z]F=a Z^{m}-b \in k[X, Y, Z] be such that a(!=0),b in k[X,Y],m >= 2a(\neq 0), b \in k[X, Y], m \geq 2 and p∤mp \nmid m. Suppose that k[X,Y,Z]//(F)=k^([2])k[X, Y, Z] /(F)=k^{[2]}. Then there exists X_(0)in k[X,Y]X_{0} \in k[X, Y] such that k[X,Y]=k[X_(0),b]k[X, Y]=k\left[X_{0}, b\right] with a in k[X_(0)]a \in k\left[X_{0}\right] and k[X,Y,Z]=k[X, Y, Z]=k[F,Z,X_(0)]k\left[F, Z, X_{0}\right].
The condition that p∤mp \nmid m is necessary in Theorem 5.2 (cf. [28, REMARK 4.6]).
Most of the above cases are covered by the following generalization due to Russell and Sathaye [76, THEOREM 3.6]:
Theorem 5.3. Let kk be a field of characteristic zero and let
F=a_(m)Z^(m)+a_(m-1)Z^(m-1)+cdots+a_(1)Z+a_(0)in k[X,Y,Z]F=a_{m} Z^{m}+a_{m-1} Z^{m-1}+\cdots+a_{1} Z+a_{0} \in k[X, Y, Z]
where a_(0),dots,a_(m)in k[X,Y]a_{0}, \ldots, a_{m} \in k[X, Y] are such that GCD(a_(1),dots,a_(m))!in k\operatorname{GCD}\left(a_{1}, \ldots, a_{m}\right) \notin k. Suppose that
k[X,Y,Z]//(F)=k^([2])k[X, Y, Z] /(F)=k^{[2]}
Then there exists X_(0)in k[X,Y]X_{0} \in k[X, Y] such that k[X,Y]=k[X_(0),b]k[X, Y]=k\left[X_{0}, b\right] with a_(m)in k[X_(0)]a_{m} \in k\left[X_{0}\right]. Further, k[X,Y,Z]=k[F]^([2])k[X, Y, Z]=k[F]^{[2]}.
Thus, for k[X,Y,Z]k[X, Y, Z], the Abhyankar-Sathaye conjecture remains open for the case when GCD(a_(1),dots,a_(m))=1\operatorname{GCD}\left(a_{1}, \ldots, a_{m}\right)=1.
A common theme in most of the partial results proved in the Abhyankar-Sathaye conjecture for k[X,Y,Z]k[X, Y, Z] is that, if FF is considered as a polynomial in ZZ, then the coordinates of k[X,Y]k[X, Y] can be so chosen that the coefficient of ZZ becomes a polynomial in XX. The Abhyankar-Sathaye conjecture for k[X,Y,Z]k[X, Y, Z] can now be split into two parts.
Question 4A. Let kk be a field of characteristic zero and let
F=a_(m)Z^(m)+a_(m-1)Z^(m-1)+cdots+a_(1)Z+a_(0)in k[X,Y,Z]F=a_{m} Z^{m}+a_{m-1} Z^{m-1}+\cdots+a_{1} Z+a_{0} \in k[X, Y, Z]
where a_(0),dots,a_(m)in k[X,Y]a_{0}, \ldots, a_{m} \in k[X, Y]. Suppose that k[X,Y,Z]//(F)=k^([2])k[X, Y, Z] /(F)=k^{[2]}. Does there exist X_(0)in k[X,Y]X_{0} \in k[X, Y] such that k[X,Y]=k[X_(0)]^([1])k[X, Y]=k\left[X_{0}\right]^{[1]} with a_(m)in k[X_(0)]a_{m} \in k\left[X_{0}\right] ?
Question 4B. Let kk be a field of characteristic zero and suppose
F=a_(m)(X)Z^(m)+a_(m-1)Z^(m-1)+cdots+a_(1)Z+a_(0)in k[X,Y,Z]F=a_{m}(X) Z^{m}+a_{m-1} Z^{m-1}+\cdots+a_{1} Z+a_{0} \in k[X, Y, Z]
where a_(0),dots,a_(m-1)in k[X,Y]a_{0}, \ldots, a_{m-1} \in k[X, Y] and a_(m)in k[X]a_{m} \in k[X]. Suppose that k[X,Y,Z]//(F)=k^([2])k[X, Y, Z] /(F)=k^{[2]}. Does this imply that k[X,Y,Z]=k[F]^([2])k[X, Y, Z]=k[F]^{[2]} ?
Sangines Garcia in his PhD thesis [78] answered Question 4A affirmatively for the case m=2m=2. In [21], Bhatwadekar and the author have given an alternative proof of this result of Garcia.
When kk is any field, as a partial generalization of Theorem 5.1 and Question 4B in four variables, the author proved the Abhyankar-Sathaye conjecture for a polynomial FF of the form X^(m)Y-F(X,Z,T)in k[X,Y,Z,T]X^{m} Y-F(X, Z, T) \in k[X, Y, Z, T]. This was one of the consequences of her general investigation on the ZCP [46]. In the process, she related it with other central problems on affine spaces like the affine fibration problem and the ZCP. The author has proved equivalence of ten statements, some of which involve an invariant introduced by Derksen, which is called the Derksen invariant.
The Derksen invariant of an integral domain BB, denoted by DK(B)\mathrm{DK}(B), is defined as the smallest subring of BB generated by the kernel of DD, where DD varies over the set of all locally nilpotent derivations of BB.
Theorem 5.4. Let kk be a field of any characteristic and AA an integral domain defined by
A=k[X,Y,Z,T]//(X^(m)Y-F(X,Z,T)),quad" where "m > 1A=k[X, Y, Z, T] /\left(X^{m} Y-F(X, Z, T)\right), \quad \text { where } m>1
Let x,y,zx, y, z, and tt denote, respectively, the images of X,Y,ZX, Y, Z, and TT in AA. Set f(Z,T):=f(Z, T):=F(0,Z,T)F(0, Z, T) and G:=X^(m)Y-F(X,Z,T)G:=X^{m} Y-F(X, Z, T). Then the following statements are equivalent:
(i) k[X,Y,Z,T]=k[X,G]^([2])k[X, Y, Z, T]=k[X, G]^{[2]}.
(ii) k[X,Y,Z,T]=k[G]^([3])k[X, Y, Z, T]=k[G]^{[3]}.
(iii) A=k[x]^([2])A=k[x]^{[2]}.
(iv) A=k^([3])A=k^{[3]}.
(v) quadA^([â„“])~=_(k)k^([â„“+3])\quad A^{[\ell]} \cong_{k} k^{[\ell+3]} for some integer â„“ >= 0\ell \geq 0 and DK(A)!=k[x,z,t]\mathrm{DK}(A) \neq k[x, z, t].
(vi) quad A\quad A is an A^(2)\mathbb{A}^{2}-fibration over k[x]k[x] and DK(A)!=k[x,z,t]\mathrm{DK}(A) \neq k[x, z, t].
(vii) AA is geometrically factorial over k,DK(A)!=k[x,z,t]k, \operatorname{DK}(A) \neq k[x, z, t] and the canonical map k^(**)rarrK_(1)(A)k^{*} \rightarrow K_{1}(A) (induced by the inclusion k↪Ak \hookrightarrow A ) is an isomorphism.
(viii) A is geometrically factorial over k,DK(A)!=k[x,z,t]k, \operatorname{DK}(A) \neq k[x, z, t] and (A//xA)^(**)=k^(**)(A / x A)^{*}=k^{*}
(ix) k[Z,T]=k[f]^([1])k[Z, T]=k[f]^{[1]}.
(x) k[Z,T]//(f)=k^([1])k[Z, T] /(f)=k^{[1]} and DK(A)!=k[x,z,t]\operatorname{DK}(A) \neq k[x, z, t].
The equivalence of (ii) and (iv) provides an answer to Question 4 for the special case of the polynomial X^(m)Y-F(X,Z,T)X^{m} Y-F(X, Z, T). The equivalence of (i) and (iii) provides an answer
to a special case of Question 4^(')4^{\prime} (stated below) for the ring R=k[x]R=k[x]. The equivalence of (iii) and (vi) answers Question 3 in a special situation. For more discussions, see [48].
In a remarkable paper Kaliman proved the following result over the field of complex numbers [56]. Later, Daigle and Kaliman extended it over any field kk of characteristic zero [25].
Theorem 5.5. Let kk be a field of characteristic zero. Let F in k[X,Y,Z]F \in k[X, Y, Z] be such that k[X,Y,Z]//(F-lambda)=k^([2])k[X, Y, Z] /(F-\lambda)=k^{[2]} for almost every lambda in k\lambda \in k. Then k[X,Y,Z]=k[F]^([2])k[X, Y, Z]=k[F]^{[2]}.
A general version of Question 4 can be asked as:
Question 4^(')4^{\prime}. Let RR be a ring and f in A=R^([n])f \in A=R^{[n]} for some integer n >= 2n \geq 2. Suppose
A//(f)~=R^([n-1])A /(f) \cong R^{[n-1]}
Does this imply that A=R[f]^([n-1])A=R[f]^{[n-1]}, i.e., is ff a coordinate in AA ?
There have been affirmative answers to Question 4^(')4^{\prime} in special cases by Bhatwadekar, Dutta, and Das [11,13, 28]. Bhatwadekar and Dutta had considered linear planes, i.e., polynomials FF of the form aZ-ba Z-b, where a,b in R[X,Y]a, b \in R[X, Y] over a discrete valuation ring RR and proved that special cases of the linear planes are actually variables. Bhatwadekar-Dutta have also shown [12] that a negative answer to Question 4^(')4^{\prime} in the case when n=3n=3 and RR is a discrete valuation ring containing Q\mathbb{Q} will give a negative answer to the affine fibration problem (Question 3 (i)) for the case n=2n=2 and d=2d=2. An example of a case of linear planes which remains unsolved is discussed in Section 7.
6. A^(n)\mathbb{A}^{\boldsymbol{n}}-FORMS
Let AA be an algebra over a field kk. We say that AA is an A^(n)\mathbb{A}^{n}-form over kk if Aox_(k)L=L^([n])A \otimes_{k} L=L^{[n]} for some finite algebraic extension LL of kk. Let AA be an A^(n)\mathbb{A}^{n}-form over a field kk.
When n=1n=1, it is well known that if L|_(k)\left.L\right|_{k} is a separable extension, then A=k^([1])A=k^{[1]} (i.e., trivial) and that if L|_(k)\left.L\right|_{k} is purely inseparable then AA need not be k^([1])k^{[1]}. An extensive study of such purely inseparable algebras was made by Asanuma in [8]. Over any field of positive characteristic, the nontrivial purely inseparable A^(1)\mathbb{A}^{1}-forms can be used to give examples of nontrivial A^(n)\mathbb{A}^{n}-forms for any integer n > 1n>1.
When n=2n=2 and L|_(k)\left.L\right|_{k} is a separable extension, then Kambayashi established that A=k^([2])A=k^{[2]} [57]. However, the problem of existence of nontrivial separable A^(3)\mathbb{A}^{3}-forms is open in general. A few recent partial results on the triviality of separable A^(3)\mathbb{A}^{3}-forms are mentioned below.
Let AA be an A^(3)\mathbb{A}^{3}-form over a field kk of characteristic zero and bar(k)\bar{k} be an algebraic closure of kk. Then A=k^([3])A=k^{[3]} if it satisfies any one of the following:
(1) AA admits a fixed point free locally nilpotent derivation DD (Daigle and Kaliman [25, COROLLARY 3.3]).
(2) AA contains an element ff which is a coordinate of Aox_(k) bar(k)A \otimes_{k} \bar{k} (Daigle and Kaliman [25, PROPOSITION 4.9]).
(3) AA admits an effective action of a reductive algebraic kk-group of positive dimension (Koras and Russell [61, THEOREM c]).
(4) AA admits either a fixed point free locally nilpotent derivation or a nonconfluent action of a unipotent group of dimension two (Gurjar, Masuda, and Miyanishi [51]).
(5) AA admits a locally nilpotent derivation DD such that rk(D ox1_( bar(k))) <= 2\operatorname{rk}\left(D \otimes 1_{\bar{k}}\right) \leq 2 (Dutta, Gupta, and Lahiri [39]).
Now let RR be a ring containing a field kk. An RR-algebra AA is said to be an A^(n)\mathbb{A}^{n}-form over RR with respect to kk if Aox_(k) bar(k)=(Rox_(k)( bar(k)))^([n])A \otimes_{k} \bar{k}=\left(R \otimes_{k} \bar{k}\right)^{[n]}, where bar(k)\bar{k} denotes the algebraic closure of kk. A few results on triviality of separable A^(n)A^{n}-forms over a ring RR are listed below.
Let AA be an A^(n)\mathbb{A}^{n}-form over a ring RR containing a field kk of characteristic 0 . Then:
(1) If n=1n=1, then AA is isomorphic to the symmetric algebra of a finitely generated rank one projective module over RR [35, THEOREM 7].
(2) If n=2n=2 and RR is a PID containing Q\mathbb{Q}, then A=R^([2])A=R^{[2]} [35, REMARK 8].
(3) If n=2n=2, then AA is an A^(2)\mathbb{A}^{2}-fibration over RR.
(4) If n=2n=2 and RR is a one-dimensional Noetherian domain, then there exists a finitely generated rank-one projective RR-module QQ such that A~=(Sym_(R)(Q))^([1])A \cong\left(\operatorname{Sym}_{R}(Q)\right)^{[1]} [39, THEOREM 3.7].
(5) If n=2n=2 and AA admits has a fixed point free locally nilpotent RR-derivation over any ring RR, then there exists a finitely generated rank one projective RR-module QQ such that A~=(Sym_(R)(Q))^([1])A \cong\left(\operatorname{Sym}_{R}(Q)\right)^{[1]} [39, THEOREM 3.8].
The result (3) above shows that an affirmative answer to the A^(2)\mathbb{A}^{2}-fibration problem (Question 3 (i)) will ensure an affirmative answer to the problem of A^(2)\mathbb{A}^{2}-forms over general rings. Over a field FF of any characteristic, Das has shown [27] that any factorial A^(1)\mathbb{A}^{1}-form AA over a ring RR containing FF is trivial if there exists a retraction map from AA to RR.
We cannot say much about A^(3)\mathbb{A}^{3}-forms over general rings till the time we solve it over fields.
7. AN EXAMPLE OF BHATWADEKAR AND DUTTA
The following example arose from the study of linear planes over a discrete valuation ring by Bhatwadekar and Dutta [12]. Question 5 stated below is an open problem for at least three decades. Let
A=C[T,X,Y,Z]quad" and "quad R=C[T,F]sub AA=\mathbb{C}[T, X, Y, Z] \quad \text { and } \quad R=\mathbb{C}[T, F] \subset A
where F=TX^(2)Z+X+T^(2)Y+TXY^(2)F=T X^{2} Z+X+T^{2} Y+T X Y^{2}.
Let
{:[P:=XZ+Y^(2)],[G:=TY+XP]:}\begin{aligned}
& P:=X Z+Y^{2} \\
& G:=T Y+X P
\end{aligned}
and
H:=T^(2)Z-2TYP-XP^(2)H:=T^{2} Z-2 T Y P-X P^{2}
Then, we can see that
XH+G^(2)=T^(2)PX H+G^{2}=T^{2} P
and F=X+TGF=X+T G. Clearly, C[T,T^(-1)][F,G,H]subeC[T,T^(-1)][X,Y,Z]\mathbb{C}\left[T, T^{-1}\right][F, G, H] \subseteq \mathbb{C}\left[T, T^{-1}\right][X, Y, Z].
Then the following statements hold:
(i) C[T,T^(-1)][X,Y,Z]=C[T,T^(-1),F,G,H]=C[T,T^(-1)][F]^([2])\mathbb{C}\left[T, T^{-1}\right][X, Y, Z]=\mathbb{C}\left[T, T^{-1}, F, G, H\right]=\mathbb{C}\left[T, T^{-1}\right][F]^{[2]}.
(ii) C[T,X,Y,Z]\mathbb{C}[T, X, Y, Z] is an A^(2)\mathbb{A}^{2}-fibration over C[T,F]\mathbb{C}[T, F].
(iii) C[T,X,Y,Z]^([1])=C[T,F]^([3])\mathbb{C}[T, X, Y, Z]^{[1]}=\mathbb{C}[T, F]^{[3]}
(iv) C[T,X,Y,Z]//(F)=C[T]^([2])=C^([3])\mathbb{C}[T, X, Y, Z] /(F)=\mathbb{C}[T]^{[2]}=\mathbb{C}^{[3]}.
(v) C[T,X,Y,Z]//(F-f(T))=C[T]^([2])\mathbb{C}[T, X, Y, Z] /(F-f(T))=\mathbb{C}[T]^{[2]} for every polynomial f(T)inC[T]f(T) \in \mathbb{C}[T].
(vii) For any u in(T,F)R,A[1//u]=R[1//u]^([2])u \in(T, F) R, A[1 / u]=R[1 / u]^{[2]}, i.e., C[T,X,Y,Z][1//u]=\mathbb{C}[T, X, Y, Z][1 / u]=C[T,F,1//u]^([2])\mathbb{C}[T, F, 1 / u]^{[2]}.
Question 5.
(a) Is A=C[T,F]^([2])(=R^([2]))A=\mathbb{C}[T, F]^{[2]}\left(=R^{[2]}\right) ?
(b) At least is A=C[F]^([3])A=\mathbb{C}[F]^{[3]} ?
If the answer is "No" to (a), then it is a counterexample to the following problems:
(1) A^(2)\mathbb{A}^{2}-fibration Problem over C^([2])\mathbb{C}^{[2]} by (ii).
(2) Cancellation Problem over C^([2])\mathbb{C}^{[2]} by (iii).
(3) Epimorphism problem over the ring C[T]\mathbb{C}[T] (see Question 4^(')4^{\prime} ) by (iv).
If the answer is "No" to (b) and hence to (a), then it is a counterexample also to the Epimorphism Problem for C^([4])rarrC^([3])\mathbb{C}^{[4]} \rightarrow \mathbb{C}^{[3]}.
Z=(H+2TYP+XP^(2))//T^(2)Z=\left(H+2 T Y P+X P^{2}\right) / T^{2}
and hence equation (1) follows.
(ii) Clearly, AA is a finitely generated RR-algebra. It can be shown by standard arguments that AA is a flat RR-algebra [66, THEOREM 20.H]. We now show that Aox_(R)k(p)=k(p)^([2])A \otimes_{R} k(p)=k(p)^{[2]} for every prime ideal pp of RR. We note that F-X in TAF-X \in T A and hence the image of FF in A//TAA / T A is same as that of XX. Now let pp be a prime ideal of RR. Then either T in pT \in p or T!in pT \notin p. If T in pT \in p, then Aox_(R)k(p)=k(p)[Y,Z]=k(p)^([2])A \otimes_{R} k(p)=k(p)[Y, Z]=k(p)^{[2]}. If T!in pT \notin p, then image of TT in k(p)k(p) is a unit and the result follows from (i).
(iii) Let D=A[W]=C[T,X,Y,Z,W]=C^([5])D=A[W]=\mathbb{C}[T, X, Y, Z, W]=\mathbb{C}^{[5]}. We shall show that
and that C[T,F,G_(2),H_(1),W_(2)]sube D\mathbb{C}\left[T, F, G_{2}, H_{1}, W_{2}\right] \subseteq D. Let D//TD=C[x,y,z,w]D / T D=\mathbb{C}[x, y, z, w], where x,y,z,wx, y, z, w denote the images of X,Y,Z,WX, Y, Z, W in D//TDD / T D. We now show that D subeC[T,F,G_(2),H_(1),W_(2)]D \subseteq \mathbb{C}\left[T, F, G_{2}, H_{1}, W_{2}\right]. For this, it is enough to show that the kernel of the natural map phi:C[T,F,G_(2),H_(1),W_(2)]rarr D//TD\phi: \mathbb{C}\left[T, F, G_{2}, H_{1}, W_{2}\right] \rightarrow D / T D is generated by TT. We note that the image of phi\phi is
C[x,y-xw,z+2yw-xw^(2),w+2p(y-xw-xp^(2))(y-xw)-xp(z+2yw-xw^(2))]\mathbb{C}\left[x, y-x w, z+2 y w-x w^{2}, w+2 p\left(y-x w-x p^{2}\right)(y-x w)-x p\left(z+2 y w-x w^{2}\right)\right],
which is of transcendence degree 4 over C\mathbb{C}. Hence the kernel of phi\phi is a prime ideal of height one and is generated by TT. Therefore, D=C[T,F,G_(2),H_(1),W_(2)]D=\mathbb{C}\left[T, F, G_{2}, H_{1}, W_{2}\right].
(iv)-(v) Let B=C[T,X,Y,Z]//(F-f(T))B=\mathbb{C}[T, X, Y, Z] /(F-f(T)) for some polynomial f inC[T]f \in \mathbb{C}[T] and S=C[T]S=\mathbb{C}[T]. By (ii), it follows that BB is an A^(2)\mathbb{A}^{2}-fibration over SS. Hence, by Sathaye's theorem [81], BB is locally a polynomial ring over SS and hence by Theorem 4.1, BB is a polynomial ring over SS.
(vi) Let H_(1):=(FH+G^(2))/(T)H_{1}:=\frac{F H+G^{2}}{T}. Then
H_(1)=((X+TG)(T^(2)Z-2TYP-XP^(2))+(TY+XP)^(2))/(T)=TP+GHH_{1}=\frac{(X+T G)\left(T^{2} Z-2 T Y P-X P^{2}\right)+(T Y+X P)^{2}}{T}=T P+G H
Let H_(2):=(FH_(1)+G^(3))/(T)H_{2}:=\frac{F H_{1}+G^{3}}{T}. Then
{:[H_(2)=((X+TG)(TP+GH)+G^(3))/(T)],[=(T(G^(2)H+TGP+XP)+G(XH+G^(2)))/(T)],[=(T(G^(2)H+TGP+XP)+GT^(2)P)/(T)],[=G^(2)H+XP+2TGP]:}\begin{aligned}
H_{2} & =\frac{(X+T G)(T P+G H)+G^{3}}{T} \\
& =\frac{T\left(G^{2} H+T G P+X P\right)+G\left(X H+G^{2}\right)}{T} \\
& =\frac{T\left(G^{2} H+T G P+X P\right)+G T^{2} P}{T} \\
& =G^{2} H+X P+2 T G P
\end{aligned}
Let H_(3):=(F(H_(2)-G)+G^(4))/(T)H_{3}:=\frac{F\left(H_{2}-G\right)+G^{4}}{T}. Then
{:[H_(3)=(F(G^(2)H+XP+2TGP-XP-TY)+G^(4))/(T)],[=(F(2TGP-TY)+G^(2)(FH+G^(2)))/(T)],[=(TF(2GP-Y)+TH_(1)G^(2))/(T)],[=F(2GP-Y)+H_(1)G^(2)]:}\begin{aligned}
H_{3} & =\frac{F\left(G^{2} H+X P+2 T G P-X P-T Y\right)+G^{4}}{T} \\
& =\frac{F(2 T G P-T Y)+G^{2}\left(F H+G^{2}\right)}{T} \\
& =\frac{T F(2 G P-Y)+T H_{1} G^{2}}{T} \\
& =F(2 G P-Y)+H_{1} G^{2}
\end{aligned}
and that the image of C[T,F,F^(-1),G,H_(2)]\mathbb{C}\left[T, F, F^{-1}, G, H_{2}\right] in A[F^(-1)]//TA[F^(-1)]A\left[F^{-1}\right] / T A\left[F^{-1}\right] is of transcendence degree 3 . Hence A[F^(-1)]=C[T,F,F^(-1),G,H_(3)]=C[T,F,F^(-1),G]^([1])A\left[F^{-1}\right]=\mathbb{C}\left[T, F, F^{-1}, G, H_{3}\right]=\mathbb{C}\left[T, F, F^{-1}, G\right]^{[1]}.
(vii) Let mm be any maximal ideal of RR other than (T,F)(T, F). Then either T!in mT \notin m or F!in mF \notin m. Thus, in either case, from (i) and (vi), we have A_(m)=R_(m)^([2])A_{m}=R_{m}^{[2]}.
Let u in(T,F)Ru \in(T, F) R. Then a maximal ideal of R[1//u]R[1 / u] is an extension of a maximal ideal of RR other than (T,F)R(T, F) R. Hence A[1//u]A[1 / u] is a locally polynomial ring in two variables over R[1//u]R[1 / u]. Further any projective module over R[1//u]R[1 / u] is free. Thus, by Theorem 4.1 , we have A[1//u]=R[1//u]^([2])A[1 / u]=R[1 / u]^{[2]}.
ACKNOWLEDGMENTS
The author thanks Professor Amartya Kumar Dutta for introducing and guiding her to this world of affine algebraic geometry. The author also thanks him for carefully going through this draft and improving the exposition.
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The formal model of semi-infinite flag manifold is a variant of an affine flag variety that was recognized from the 1980s but not studied extensively until the late 2010s. In this note, we exhibit constructions and ideas appearing in our recent study of the formal model of semi-infinite flag manifold of a simple algebraic group. Our results have some implications to the theory of rational maps from a projective line to partial flag manifolds, and also on the structures of quantum cohomologies and quantum KK-groups of partial flag manifolds.
Semi-infinite flag manifold, quasi-map space, quantum K-group, Kac-Moody group, affine
Lie algebra, global Weyl module
1. INTRODUCTION
Compact complex-analytic spaces that admit homogeneous Lie group actions are quite rare in nature, and their classification reduces into three primitive classes: finite groups, tori, and (partial) flag manifolds. The first have discrete topology and the role of geometric consideration is rather small, in general. The second, particularly those admit polarizations, offer a major subject known as abelian varieties. The third, the (partial) flag manifolds of compact simple Lie groups, are ubiquitous in representation theory of semisimple algebraic groups and quantum groups. By the universal nature of general linear groups, flag manifolds of unitary groups are extensively studied from the geometric perspective.
In representation-theoretic considerations, we usually consider flag manifolds as projective algebraic varieties defined over an algebraically closed field (that form a family over Spec Z\operatorname{Spec} \mathbb{Z} ). This definition naturally extends to an arbitrary Kac-Moody setting, but the resulting objects have at least two variants, thin flag varieties and thick flag manifolds (defined by Kac-Peterson [75] and Kashiwara [40], respectively). In case the Kac-Moody group is of affine type, we have a loop realization of the corresponding Kac-Moody group, essentially identifying the corresponding group with the set of k((z))\mathbb{k}((z))-valued points of a simple algebraic group over a field k\mathbb{k}. This motivates us to consider yet other versions of flag manifolds of affine type that can be understood as an enhancement of arc schemes of usual flag manifolds. These are the semi-infinite flag manifolds that originate from the ideas of Lusztig [63, $11] and Drinfeld [22]. Lusztig's original idea is to construct varieties that naturally encode representation theory of simple algebraic groups over finite fields. The Lusztig program (see, e.g., [44,63][44,63] ) adds representation theory of quantum groups at roots of unity and representation theory of affine Lie algebras at negative rational levels into the picture, and Feigin-Frenkel [19] put representation theory of affine Lie algebras at the critical level into the picture. The semi-infinite flag manifolds itself have two realizations, that we refer to as the ind-model and the formal model. The geometry of the ind-model of semi-infinite flag manifolds, also known as the space of quasimaps from a projective line to a flag manifold, was studied extensively by Braverman, Finkelberg, Mirković, and their collaborators (see [8,18,21,22][8,18,21,22] ).
One instance of the ind-model of semi-infinite flag manifold is the space of principal bundles on an algebraic curve equipped with some reduction. This interpretation realizes some portion of the above representation-theoretic expectations [2,31]. The formal model of semi-infinite flag manifolds is expected to add a concrete understanding of related representation-theoretic patterns [19,22,63][19,22,63]. Unfortunately, such an idea needs to be polished as its implementation faces difficulty due to its essential infinite-dimensionality. This forces us to employ affine Grassmannians instead of semi-infinite flag manifolds in some cases (see [26,30,78][26,30,78] ) at the moment, that is possible by some tight connections [27,70][27,70].
Meanwhile, it is realized that the semi-infinite flag manifold is a version of the loop space of a flag manifold, and hence it is related to its quantum cohomology [32]. In fact, the ind-model of a semi-infinite flag manifold offers a description of the quantum KK-theoretic JJ-function of a flag manifold [9] that encodes its small quantum KK-group.
In both contexts of the above two paragraphs, the Peterson isomorphism [59, 74], that connects the quantum cohomology of a flag manifold with the homology of an affine Grassmannian, should admit an interpretation using a semi-infinite flag manifold. However, such an interpretation is not known today (though we have Corollary 7.3).
The main goal of this note is to explain a realization of the formal model of semiinfinite flag manifold [46,50,52], that is reminiscent to the classical description of the original flag manifolds. Our realization is supported by recent developments in representation theory of affine Lie algebras [14,15,51][14,15,51], that is also reminiscent to the representation theory of simple Lie algebras. It turns out that the study of the formal model of the semi-infinite flag manifold has implications to the corresponding ind-model [50], as well as the study of quantum KK-groups of partial flag manifolds and the KK-groups of affine Grassmannians [45, 47, 48]. This includes an interpretation (and a proof) of the KK-theoretic analogue of the Peterson isomorphism using semi-infinite flag manifolds (Theorem 8.2).
The results presented here describe the formal model of semi-infinite flag manifolds in a down-to-earth fashion, and also provide first nontrivial conclusions deduced from them. However, we have not yet reached our primary goal to understand representation theory from this perspective in a satisfactory fashion. We hope to improve this situation in the near future.
The organization of this note is as follows: We first recall the construction of flag manifolds that is parallel to our later construction in Section 2. We explain the role of quantum groups in the structure theory of Kac-Moody algebras and exhibit two versions of flag varieties of Kac-Moody groups in Section 3. In Section 4, we exhibit some representation theory of affine Lie algebras. Based on it, we explain our construction of the formal model of semi-infinite flag manifolds in Section 5. This enables us to present our idea on the Frobenius splitting of semi-infinite flag manifolds in Section 6. We explain the connection between its Richardson varieties and quasimap spaces in Section 7, and explain how they fit into the study of quantum cohomology of flag manifolds. We exhibit the KK-theoretic Peterson isomorphism in Section 8. We discuss the functoriality of the quantum KK-groups of partial flag manifolds in Section 9. We finish this note by discussing some perspectives in Section 10.
We assume that every field k\mathbb{k} has characteristic !=2\neq 2. A variety is some algebraicgeometric object that admits singularity, and a manifold is a variety that is supposed to be smooth in some sense. An algebraic variety is a separated scheme of finite type defined over a field (i.e., our variety is not necessarily irreducible or reduced). We set N:=Z_( >= 0)\mathbb{N}:=\mathbb{Z}_{\geq 0}.
2. FLAG MANIFOLDS VIA REPRESENTATION THEORY
Let GG be a simply connected semisimple algebraic group over an algebraically closed field k\mathbb{k}. Let T sub BT \subset B be its maximal torus and a Borel subgroup (maximal solvable subgroup). Let W(=N_(G)(T)//T)W\left(=N_{G}(T) / T\right) be the Weyl group of GG. Let X\mathbb{X} be the set of onedimensional rational TT-characters (the set of TT-weights), that admits a natural WW-action. We set X_(+):=sum_(i=1)^(r)N_(i)\mathbb{X}_{+}:=\sum_{i=1}^{r} \mathbb{N}_{i}, where Ï–_(1),dots,Ï–_(r)inX\varpi_{1}, \ldots, \varpi_{r} \in \mathbb{X} are fundamental weights with respect to BB. The set of isomorphism classes of irreducible rational representations {L(lambda)}_(lambda)\{L(\lambda)\}_{\lambda} of GG is labeled by X_(+)\mathbb{X}_{+}in such a way that each L(lambda)L(\lambda) contains a unique (up to scalar) BB-eigenvector v_(lambda)\mathbf{v}_{\lambda} with its TT-weight lambda\lambda. We refer lambda(inX_(+))\lambda\left(\in \mathbb{X}_{+}\right)as the highest weight of L(lambda)L(\lambda). The flag manifold B:=G//B\mathscr{B}:=G / B of GG is the maximal GG-homogeneous space that is projective.
In case k=C\mathbb{k}=\mathbb{C}, we have
B=(y\\E)//T\mathscr{B}=(y \backslash E) / T
where yy is an affine algebraic variety with (G xx T)(G \times T)-action whose ring C[y]\mathbb{C}[y] of regular functions is written as
{:(2.1)C[y]~=_(lambda inX_(+))L(lambda)^(**)quad(" as "G xx T"-modules "):}\begin{equation*}
\mathbb{C}[y] \cong \underset{\lambda \in \mathbb{X}_{+}}{ } L(\lambda)^{*} \quad(\text { as } G \times T \text {-modules }) \tag{2.1}
\end{equation*}
and E sub yE \subset y is the locus where the TT-action is not free. Here, the GG-action on C[y]\mathbb{C}[y] is the natural actions on L(lambda)L(\lambda), and the TT-action on C[y]\mathbb{C}[y] comes from the grading X_(+)subX\mathbb{X}_{+} \subset \mathbb{X} in the RHS of (2.1). These data, together with the condition E!=yE \neq y, essentially determine C[y]\mathbb{C}[y] as C\mathbb{C}-algebras generated by L(Ï–_(i))^(**)L\left(\varpi_{i}\right)^{*} for 1 <= i <= r1 \leq i \leq r. Consider a point x_(0)in yx_{0} \in y given by {v_(lambda)}lambda\left\{\mathbf{v}_{\lambda}\right\} \lambda, seen as linear maps on {L(lambda)^(**)}_(lambda)\left\{L(\lambda)^{*}\right\}_{\lambda}. The image [x_(0)]\left[x_{0}\right] of this point x_(0)x_{0} has its GG-stabilizer equal to BB. This induces an inclusion
G//B↪Bsubprod_(i=1)^(r)P(L(ϖ_(i)))G / B \hookrightarrow \mathscr{B} \subset \prod_{i=1}^{r} \mathbb{P}\left(L\left(\varpi_{i}\right)\right)
induced from B//B|->[x_(0)]B / B \mapsto\left[x_{0}\right] by the GG-action. (One needs additional representation-theoretic analysis to conclude G//B~=BG / B \cong \mathscr{B}.) This consideration transfers all geometric statements relevant to B\mathscr{B} to algebraic statements on the space in (2.1) in principle, but most of the geometric results on B\mathscr{B} and its subvarieties were proved for the first time by other methods (see, e.g., [56]).
Note that the vector space (2.1) does not acquire the structure of a ring when char k=\mathbb{k}=p > 0p>0. The reason is that we do not have a map L(lambda)^(**)ox L(mu)^(**)rarr L(lambda+mu)^(**)L(\lambda)^{*} \otimes L(\mu)^{*} \rightarrow L(\lambda+\mu)^{*}, or equivalently, L(lambda+mu)rarr L(lambda)ox L(mu)L(\lambda+\mu) \rightarrow L(\lambda) \otimes L(\mu) for general lambda,mu inX_(+)\lambda, \mu \in \mathbb{X}_{+}. One way to improve the situation is to replace {L(lambda)}_(lambda inX_(+))\{L(\lambda)\}_{\lambda \in \mathbb{X}_{+}}with a suitable family of modules {Y(lambda)}_(lambda inX_(+))\{Y(\lambda)\}_{\lambda \in \mathbb{X}_{+}}with larger members such that the GG-module map
exists uniquely (up to constant) for every lambda,mu inX_(+)\lambda, \mu \in \mathbb{X}_{+}. It yields an analogous ring of (2.1) that should be closely related to B\mathscr{B}. A standard choice of Y(lambda)(lambda inX_(+))Y(\lambda)\left(\lambda \in \mathbb{X}_{+}\right)is the Weyl module V(lambda)V(\lambda) of GG, that is, the projective cover of L(lambda)L(\lambda) in the categories of rational GG-modules whose composition factors are in {L(mu)}_(lambda >= mu inX_(+))\{L(\mu)\}_{\lambda \geq \mu \in \mathbb{X}_{+}}, where >=\geqis the dominance ordering on X\mathbb{X}. This produces B\mathscr{B} for all characteristics.
Theorem 2.1 (Orthogonality of Weyl modules, [36, II $4.13]). For each lambda,mu inX_(+)\lambda, \mu \in \mathbb{X}_{+}, we have
Note that L(lambda)=V(lambda)L(\lambda)=V(\lambda) for char k=0\mathbb{k}=0 by the semisimplicity of representations, and hence Theorem 2.1 is Schur's lemma in such a case. As V(lambda)=kox_(Z)V_(Z)(lambda)V(\lambda)=\mathbb{k} \otimes_{\mathbb{Z}} V_{\mathbb{Z}}(\lambda) holds for a collection of free Z\mathbb{Z}-modules V_(Z)(lambda)(lambda inX_(+))V_{\mathbb{Z}}(\lambda)\left(\lambda \in \mathbb{X}_{+}\right), we find that B\mathscr{B} extends to a scheme flat over Z\mathbb{Z}. Another possible choice of Y(lambda)(lambda inX_(+):}Y(\lambda)\left(\lambda \in \mathbb{X}_{+}\right.), the Verma module M(lambda)M(\lambda) of the (divided power) enveloping algebra of Lie GG, produces an open dense BB-orbit in B\mathscr{B}.
3. KAC-MOODY FLAG VARIETIES
Let us keep the setting of the previous section.
3.1. Reminder on Kac-Moody algebras and their quantum groups
Let g_(C)g_{C} be the Kac-Moody algebra associated to a symmetrizable generalized Cartan matrix ( == GCM) CC (see [38]). In case char k=0k=0, we have the notion of the highest weight integrable representations of g_(C)g_{C} parametrized by the set of dominant weights P_(+)P_{+} defined similarly to X_(+)\mathbb{X}_{+}. Let L(Lambda)L(\Lambda) denote the highest weight integrable representation of g_(C)\mathrm{g}_{C} corresponding to Lambda inP_(+)\Lambda \in P_{+}.
We have the quantum group (or the quantized enveloping algebra) U_(q)(g_(C))U_{q}\left(g_{C}\right) of g_(C)g_{C} originally defined by Drinfeld and Jimbo in the 1980s [17, 37]. It is an algebra defined over Q(q)\mathbb{Q}(q), and the specialization q|->1q \mapsto 1 recovers the universal enveloping algebra U(g_(C))U\left(\mathrm{~g}_{C}\right) of g_(C)\mathrm{g}_{C}. Kashiwara [41] and Lusztig [63] defined the canonical/global bases (of the positive/negative parts {:U_(q)^(+-)(g_(C)))\left.U_{q}^{ \pm}\left(g_{C}\right)\right) of U_(q)(g_(C))U_{q}\left(g_{C}\right) and their integrable representations that generate their Q[q]-\mathbb{Q}[q]- lattices. The construction of Lusztig [64] clarified that quantum groups are, in fact, defined over Z[q^(+-1)]\mathbb{Z}\left[q^{ \pm 1}\right] (or even over N[q^(+-1)]\mathbb{N}\left[q^{ \pm 1}\right] if one can say). In the 2010s, the categorification theorems of a quantum group and its integrable representations appeared [39,53,76,77][39,53,76,77], and there every algebra that admits a categorification has a suitable Z[q]\mathbb{Z}[q]-integral structure with distinguished bases, being the Grothendieck group of a module category of a finitely-generated graded algebras (called KLR algebras or quiver Hecke algebras). Therefore, the following is now widely recognized:
Theorem 3.1 (Lusztig [63,64,66][63,64,66] and Kashiwara [41-43]). Assume that k=C\mathbb{k}=\mathbb{C}. The (lower) global bases of U_(q)^(+-)(g_(C))U_{q}^{ \pm}\left(g_{C}\right) induce a Z\mathbb{Z}-integral form U_(Z)(g_(C))U_{\mathbb{Z}}\left(g_{C}\right) of U(g_(C))U\left(g_{C}\right) via q|->1q \mapsto 1. For each Lambda inP_(+)\Lambda \in P_{+}, we have a Z\mathbb{Z}-lattice L(Lambda)_(Z)L(\Lambda)_{\mathbb{Z}} of L(Lambda)L(\Lambda) obtained from the (lower) global base of the corresponding integrable highest weight module of U_(q)(g_(C))U_{q}\left(g_{C}\right). In addition, L(Lambda)_(Z)L(\Lambda)_{\mathbb{Z}} is generated by the U_(Z)(g_(C))U_{\mathbb{Z}}\left(\mathrm{g}_{C}\right)-action from a highest weight vector of L(Lambda)L(\Lambda).
By a specialization of L(Lambda)_(Z)L(\Lambda)_{\mathbb{Z}}, we obtain a highest weight integrable module L(Lambda)L(\Lambda) over an arbitrary field k\mathbb{k}. The module L(Lambda)L(\Lambda) is no longer irreducible when char k > 0\mathbb{k}>0 (in general), and hence it is a g_(C)\mathrm{g}_{C}-analogue of Weyl modules rather than L(lambda)L(\lambda) for GG; it is a lack of brevity of the author to choose this notation here. We close this subsection by noting that the integral forms at the end of Section 2 coincide with the integral forms in Theorem 3.1.
3.2. Thin and thick flag varieties
Presentations of the flag varieties for general Kac-Moody groups E\mathscr{E} associated to a GCM CC are similar to those in the previous section. A triangular decomposition of g_(C)g_{C} yields an analogous group â„“\ell to the Borel subgroup. Let T\mathcal{T} be a (standard) maximal torus of â„“\ell. The highest weight vector in L(Lambda)L(\Lambda) is precisely an â„“\ell-eigenvector with its T\mathcal{T}-weight Lambda\Lambda. Therefore, the construction in the previous section produces E//â„“\mathscr{E} / \mathscr{\ell} via the ring
where L(Lambda)^(**)L(\Lambda)^{*} is the vector space dual of L(Lambda)L(\Lambda), and L(Lambda)^(vv)L(\Lambda)^{\vee} is the restricted dual of L(Lambda)L(\Lambda), defined to be the direct sum of (finite-dimensional) vector space duals offered by the T\mathcal{T}-weight decomposition of L(Lambda)L(\Lambda).
In this case, both vector spaces in (3.1) are naturally rings. This corresponds to the choice of E\mathscr{E}. The former ring defines B_(C)^("thick ")=E//d[40,49,71]\mathscr{B}_{C}^{\text {thick }}=\mathscr{E} / \mathscr{d}[40,49,71] if we take E\mathscr{E} to be a version of the Kac-Moody group that is completed with respect to the opposite direction to â„“\mathscr{\ell}. (This is the maximal Kac-Moody group, but the completion is taken in the opposite way as in the literature.) The latter ring can be seen as the projective limit of finitely-generated algebras, and the union of the spectrums of these rings yields B_(C)^("thin ")=E//L\mathscr{B}_{C}^{\text {thin }}=\mathscr{E} / \mathscr{L} [56,75] if we take E\mathscr{E} as the uncompleted Kac-Moody group (the Kac-Peterson group or the minimal Kac-Moody group), or as the maximal Kac-Moody group completed with respect to the direction of â„“\ell. In other words, we have variants of flag manifolds of Kac-Moody groups associated to a GCM CC as:
The scheme B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }} is a union of infinite-dimensional affine spaces, and hence is smooth. However, B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }} is not compact in an essential way [24]. This picture is compatible with the fact that the Kac-Peterson group is defined by one-parameter generators (and relations), and hence B_(C)^("thin ")\mathscr{B}_{C}^{\text {thin }} is a union of finite-dimensional subvarieties B_(C,n)^("thin ")\mathscr{B}_{C, n}^{\text {thin }} consisting of points presented by a product of at most nn generating elements. As such, each scheme B_(C,n)^("thin ")\mathscr{B}_{C, n}^{\text {thin }} is singular, and hence B_(C)^("thin ")\mathscr{B}_{C}^{\text {thin }} is understood to be singular. In fact, it does not admit an inductive limit description by finite-dimensional smooth pieces [24].
4. GLOBAL WEYL MODULES AND THEIR PROJECTIVITY
Let us consider the untwisted affine Kac-Moody case hereafter, with the same conventions as in the previous sections. In particular, our Kac-Moody groups are extensions of the groups
G((z)):=G(k((z)))quad" and "quad G[z^(+-1)]:=G(k[z^(+-1)])G((z)):=G(\mathbb{k}((z))) \quad \text { and } \quad G\left[z^{ \pm 1}\right]:=G\left(\mathbb{k}\left[z^{ \pm 1}\right]\right)
by the loop rotation G_(m)\mathbb{G}_{m}-actions (that we denote by G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }} ) and the central extension G_(m)\mathbb{G}_{m} actions. (These correspond to the maximal/minimal realizations of the Kac-Moody groups
in the previous section.) These are not (pro-)algebraic groups, and it sometimes causes difficulty. Nevertheless, each rational representation VV of GG induces representations
V((z)):=Vox_(k)k((z))quad" and "quad V[z^(+-1)]:=Vox_(k)k[z^(+-1)]V((z)):=V \otimes_{\mathbb{k}} \mathbb{k}((z)) \quad \text { and } \quad V\left[z^{ \pm 1}\right]:=V \otimes_{\mathfrak{k}} \mathbb{k}\left[z^{ \pm 1}\right]
of G((z))G((z)) and G[z^(+-1)]G\left[z^{ \pm 1}\right], respectively. These representations are not of highest weight, but still integrable representations when we lift them to the central extensions of G((z))G((z)) and G[z^(+-1)]G\left[z^{ \pm 1}\right] by letting the center G_(m)\mathbb{G}_{m} act trivially (i.e., they are level-zero integrable representations viewed as representations of affine Lie algebras).
In addition to the TT-action, the representation V[z^(+-1)]V\left[z^{ \pm 1}\right] carries G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}-action. Let delta\delta be the degree-one character of G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}, and set q:=e^(delta)q:=e^{\delta}. By abuse of notation, we might consider q^(n)q^{n}(n inZ)(n \in \mathbb{Z}) as the functor that twists the G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}-action by degree nn. We define a graded character of a semisimple (T xxG_(m)^("rot "))\left(T \times \mathbb{G}_{m}^{\text {rot }}\right)-module UU as
Then, gch V[z^(+-1)]V\left[z^{ \pm 1}\right] makes sense as all the coefficients are in Z\mathbb{Z}. However, if we take the second symmetric power S^(2)(V[z^(+-1)])S^{2}\left(V\left[z^{ \pm 1}\right]\right) of V[z^(+-1)]V\left[z^{ \pm 1}\right] over k\mathbb{k}, then it contains an infinity as a coefficient. To avoid such a complication, we sometimes restrict ourselves to the subgroups
G[[z]]:=G(k[[z]])sub G((z))quad" and "quad G[z]:=G(k[z])sub G[z^(+-1)]G \llbracket z \rrbracket:=G(\mathbb{k} \llbracket z \rrbracket) \subset G((z)) \quad \text { and } \quad G[z]:=G(\mathbb{k}[z]) \subset G\left[z^{ \pm 1}\right]
We sometimes use the subgroup Isub G[[z]]\mathbf{I} \subset G \llbracket z \rrbracket defined by the pullback of BB under the evaluation map ev_(0):G[[z]]rarr G\mathrm{ev}_{0}: G \llbracket z \rrbracket \rightarrow G at z=0z=0. The group I\mathbf{I} is the Iwahori subgroup obtained from (the completed version of )d) d by removing G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }} and quotient out by the central extension.
By the quotient map k[z]rarrk\mathbb{k}[z] \rightarrow \mathbb{k} (and k[[z]]rarrk\mathbb{k} \llbracket z \rrbracket \rightarrow \mathbb{k} ) sending z|->0z \mapsto 0, we can regard every rational GG-module VV as a G[z]G[z]-module or a G[[z]]G \llbracket z \rrbracket-module with (trivial) G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}-action through ev_(0)\mathrm{ev}_{0}. We also have a G[[z]]G \llbracket z \rrbracket-module structure (without a G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}-action) on V[[z]]:=V oxk[[z]]V \llbracket z \rrbracket:=V \otimes \mathbb{k} \llbracket z \rrbracket that surjects onto VV.
Definition 4.1 (global Weyl modules). Let ⨀(lambda)\bigodot(\lambda) be the category of rational G[z]G[z]-modules MM that admits a decreasing filtration
M=F_(0)M supF_(1)M supF_(2)M sup cdotsquad" such that "nnn_(k >= 0)F_(k)M={0}M=F_{0} M \supset F_{1} M \supset F_{2} M \supset \cdots \quad \text { such that } \bigcap_{k \geq 0} F_{k} M=\{0\}
and each F_(k)M//F_(k-1)M(k >= 1)F_{k} M / F_{k-1} M(k \geq 1) belongs to {q^(m)L(mu)}_(m inZ,lambda >= mu inX_(+))\left\{q^{m} L(\mu)\right\}_{m \in \mathbb{Z}, \lambda \geq \mu \in \mathbb{X}_{+}}. For each lambda inX_(+)\lambda \in \mathbb{X}_{+}, we define the global Weyl module W(lambda)\mathbb{W}(\lambda) of G[z]G[z] as the projective cover of L(lambda)L(\lambda) in ⨀(lambda)\bigodot(\lambda).
Note that W(lambda)\mathbb{W}(\lambda) automatically acquires a G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}-action by its universality (as it exists).
Theorem 4.2. For each lambda inX_(+)\lambda \in \mathbb{X}_{+}with lambda=sum_(i=1)^(r)m_(i)Ï–_(i)\lambda=\sum_{i=1}^{r} m_{i} \varpi_{i}, we have
where each x_(i,1),dots,x_(i,m_(i))x_{i, 1}, \ldots, x_{i, m_{i}} is of degree one with respect to the G_(m)^(rot)\mathbb{G}_{m}^{\mathrm{rot}}-action. In addition, the action of End_(G[z])W(lambda)\operatorname{End}_{G[z]} \mathbb{W}(\lambda) on W(lambda)\mathbb{W}(\lambda) is free.
Theorem 4.2 was proved by Fourier-Littelmann [25] (for k=C\mathbb{k}=\mathbb{C} and GG of type ADE), Naoi [72] (for k=C\mathbb{k}=\mathbb{C} and GG of type BCFG), and it was transferred to char k > 0\mathbb{k}>0 in [50] using results from the global bases of quantum affine algebras [4,42][4,42].
By Theorem 4.2, we factor out the positive degree parts of End_(G[z])W(lambda)\operatorname{End}_{G[z]} \mathbb{W}(\lambda) to obtain
We call it a local Weyl module of G[z]G[z].
The following result clarifies that our global/local Weyl modules are the best possible analogues of Weyl modules for GG (see Theorem 2.1):
Theorem 4.3 (Chari-Ion [14] for char k=0\mathbb{k}=0, and [50] +epsi+\varepsilon for char k > 0\mathbb{k}>0 ). For each lambda,mu inX_(+)\lambda, \mu \in \mathbb{X}_{+}, we have
The proof of Theorem 4.3 in [50,$3.3][50, \$ 3.3] relies on the adjoint property of the Demazure functors observed in [20, PROPOSITION 5.7] and systematically utilized in [15]. The case lambda=mu^(**)\lambda=\mu^{*} and i > 1i>1 in Theorem 4.3 is not recorded in [50], and might appear elsewhere.
5. SEMI-INFINITE FLAG MANIFOLDS
We keep the setting of the previous section. In view of the projectivity of W(lambda)\mathbb{W}(\lambda) 's in ⨀(lambda)\bigodot(\lambda) 's, we find unique degree-zero G[z]G[z]-module maps
with a structure of a commutative algebra compatible with the action of G[z]><|G_(m)^("rot ")xx TG[z] \rtimes \mathbb{G}_{m}^{\text {rot }} \times T. Since the G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}-degree of R_(G)R_{G} is bounded from the above, the G[z]G[z]-action on R_(G)R_{G} automatically extends to the G[[z]]G \llbracket z \rrbracket-action. We set
Q_(G):=(Spec R_(G)\\E)//T\mathbf{Q}_{G}:=\left(\operatorname{Spec} R_{G} \backslash E\right) / T
where EE is a closed subset of Spec R_(G)R_{G} on which the TT-action is not free. Let us consider the G((z))G((z))-orbit of
viewed as a set of points, that we denote by Q_(G)\mathcal{Q}_{G}. By examining the coefficients of the defining relations of B\mathscr{B} with its k((z))\mathbb{k}((z))-valued points, we find that the intersection
defines a closed subscheme for any choice of m_(1),dots,m_(r)inZm_{1}, \ldots, m_{r} \in \mathbb{Z}. We denote this subscheme by Q_(G)(t_(beta))\mathbf{Q}_{G}\left(t_{\beta}\right), where beta=sum_(i=1)^(r)m_(i)alpha_(i)^(vv)\beta=\sum_{i=1}^{r} m_{i} \alpha_{i}^{\vee} is an element of the dual lattice (coroot lattice) X^(vv)\mathbb{X}^{\vee} of X\mathbb{X} equipped with a basis {alpha_(i)^(vv)}_(i=1)^(r)\left\{\alpha_{i}^{\vee}\right\}_{i=1}^{r} such that alpha_(i)^(vv)(Ï–_(j))=delta_(i,j)\alpha_{i}^{\vee}\left(\varpi_{j}\right)=\delta_{i, j} (i.e., alpha_(i)^(vv)\alpha_{i}^{\vee} is a simple coroot). We note that P(V(Ï–_(i))([[)z(]])z^(m_(i)))\mathbb{P}\left(V\left(\varpi_{i}\right) \llbracket z \rrbracket z^{m_{i}}\right) is a scheme, but it is not of finite type, and Q_(G)(t_(beta))\mathbf{Q}_{G}\left(t_{\beta}\right) is also of infinite type.
Lemma 5.1. We have Q_(G)(t_(beta))~=Q_(G)(t_(gamma))\mathbf{Q}_{G}\left(t_{\beta}\right) \cong \mathbf{Q}_{G}\left(t_{\gamma}\right) for each pair beta,gamma inX^(vv)\beta, \gamma \in \mathbb{X}^{\vee} as schemes equipped with G[[z]]G \llbracket z \rrbracket-actions. Hence, the union
is a pure ind-scheme of ind-infinite type equipped with the action of G[[z]]><|G_(m)^(rot)G \llbracket z \rrbracket \rtimes \mathbb{G}_{m}^{\mathrm{rot}}. Moreover, the set of G[[z]]G \llbracket z \rrbracket-orbits in Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }} is in bijection with X^(vv)\mathbb{X}^{\vee}.
In effect, we have an open dense G[[z]]G \llbracket z \rrbracket-orbit O_(G)(t_(beta))subQ_(G)(t_(beta))\mathbf{O}_{G}\left(t_{\beta}\right) \subset \mathbf{Q}_{G}\left(t_{\beta}\right) that is isomorphic to G[[z]]//(T*N[[z]])G \llbracket z \rrbracket /(T \cdot N \llbracket z \rrbracket). By the Bruhat decomposition, we divide O_(G)(t_(beta))\mathbf{O}_{G}\left(t_{\beta}\right) into the disjoint union of I\mathbf{I}-orbits as ⨆_(w in W)O(wt_(beta))\bigsqcup_{w \in W} \mathbf{O}\left(w t_{\beta}\right) such that O(t_(beta))subO_(G)(t_(beta))\mathbf{O}\left(t_{\beta}\right) \subset \mathbf{O}_{G}\left(t_{\beta}\right) is open dense. Identifying beta inXvv\beta \in \mathbb{X} \vee with t_(beta)t_{\beta}, we set W_(af):=W|><X^(vv)W_{\mathrm{af}}:=W \ltimes \mathbb{X}^{\vee}. We define
Q_(G)(w):= bar(O(w))subQ_(G)^(rat),quad w inW_(af)\mathbf{Q}_{G}(w):=\overline{\mathbf{O}(w)} \subset \mathbf{Q}_{G}^{\mathrm{rat}}, \quad w \in W_{\mathrm{af}}
The inclusion relation on {Q_(G)(w)}_(w inW_(af))\left\{\mathbf{Q}_{G}(w)\right\}_{w \in W_{\mathrm{af}}} is described by the generic Bruhat order [62]. We refer to the partial order on W_("af ")W_{\text {af }} induced from this closure ordering by <= (oo)/(2)\leq \frac{\infty}{2} as in [50,52][50,52] (there we sometimes called <= (oo)/(2)\leq \frac{\infty}{2} as the semi-infinite Bruhat order).
Theorem 5.2. The scheme Q_(G)(w)\mathbf{Q}_{G}(w) is normal for each w inW_(af)w \in W_{\mathrm{af}}. In addition, the ind-scheme Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }} is a strict ind-scheme in the sense that each inclusion is a closed immersion. The indscheme Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }} coarsely ind-represents the coset G((z))//(T*N((z)))G((z)) /(T \cdot N((z))).
The first two statements are proved in [52] when char k=0k=0. The proof valid for char k!=2\mathbb{k} \neq 2, as well as the last assertion, are contained in [50]. This last assertion says that the (ind-)scheme Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }} is the universal one that maps to every (ind-)scheme whose points yield Q_(G)Q_{G}. It follows that if we take a family {Y(lambda)}_(lambda inX_(+))\{\mathbb{Y}(\lambda)\}_{\lambda \in \mathbb{X}_{+}}instead of {W(lambda)}_(lambda inX_(+))\{\mathbb{W}(\lambda)\}_{\lambda \in \mathbb{X}_{+}}to define Q_(G)(t_(beta))\mathbf{Q}_{G}\left(t_{\beta}\right), then the corresponding coordinate ring R_(G)^(')R_{G}^{\prime} admits a map to R_(G)R_{G}. Let us point out that this can be thought of as a family version of the properties of global Weyl modules discussed in Section 4, and we indeed have several reasonable choices of {Y(lambda)}_(lambda inX_(+))\{\mathbb{Y}(\lambda)\}_{\lambda \in \mathbb{X}_{+}}other than {W(lambda)}_(lambda inX_(+))\{\mathbb{W}(\lambda)\}_{\lambda \in \mathbb{X}_{+}}including the coordinate ring of the arc scheme of G//NG / N. For simplicity, we may refer to Q_(G)(t_(0))\mathbf{Q}_{G}\left(t_{0}\right) as Q_(G)\mathbf{Q}_{G} below.
The inclusion
{:(5.4)Q_(G)subprod_(i=1)^(r)P(V(Ï–_(i))([[)z(]])):}\begin{equation*}
\mathbf{Q}_{G} \subset \prod_{i=1}^{r} \mathbb{P}\left(V\left(\varpi_{i}\right) \llbracket z \rrbracket\right) \tag{5.4}
\end{equation*}
induces a line bundle O_(Q_(G))(ϖ_(i))\mathcal{O}_{\mathbf{Q}_{G}}\left(\varpi_{i}\right) on Q_(G)\mathbf{Q}_{G}, that is, the pull-back of O(1)\mathcal{O}(1) from P(V(ϖ_(i))([[)z(]]))\mathbb{P}\left(V\left(\varpi_{i}\right) \llbracket z \rrbracket\right). By taking the tensor products, we have O_(Q_(G))(lambda):=⨂_(i=1)^(r)O_(Q_(G))(ϖ_(i))^(oxn_(i))\mathcal{O}_{\mathbf{Q}_{G}}(\lambda):=\bigotimes_{i=1}^{r} \mathcal{O}_{\mathbf{Q}_{G}}\left(\varpi_{i}\right)^{\otimes n_{i}} for lambda=sum_(i=1)^(r)n_(i)ϖ_(i)\lambda=\sum_{i=1}^{r} n_{i} \varpi_{i}(n_(i)inZ)\left(n_{i} \in \mathbb{Z}\right). By Lemma 5.1, we have O_(Q_(G)^("rat "))(lambda)(lambda inX)\mathcal{O}_{\mathbf{Q}_{G}^{\text {rat }}}(\lambda)(\lambda \in \mathbb{X}) on Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }} that yields O_(Q_(G))(lambda)\mathcal{O}_{\mathbf{Q}_{G}}(\lambda) by restriction.
Theorem 5.3 ([52] for char k=0\mathbb{k}=0, and [50] for char k!=2\mathbb{k} \neq 2 ). For each lambda inX\lambda \in \mathbb{X}, we have
The proof of Theorem 5.3 depends on the freeness of R_(G)R_{G} over an infinitely-manyvariable polynomial ring, that yields a regular sequence of infinite length. Such a situation never occur for finite type schemes, or infinite type schemes like B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}. In case G=SL(2)G=\operatorname{SL}(2), Theorem 5.3 reduces to an exercise in algebraic geometry by Q_(G)~=P(k^(2)([[)z(]]))\mathbf{Q}_{G} \cong \mathbb{P}\left(\mathbb{k}^{2} \llbracket z \rrbracket\right).
Theorem 5.3 has an ind-model counterpart proved earlier [10]. The Frobenius splitting of Q_(G)\mathbf{Q}_{G} (explained later) and Theorem 5.3 imply this ind-model counterpart. However, the author is uncertain whether [10] implies Theorem 5.3 (even in case char k=0k=0 ) since the natural ring coming from the ind-model is a completion of R_(G)R_{G}, and the completion operation of a ring loses information in general. We have an analogue of Theorem 5.3 for all I\mathbf{I}-orbit closures, proved for the ind-model in [46,50] and for the formal model in [50,52].
6. FROBENIUS SPLITTINGS
We continue to work in the setting of the previous section. We fix a prime p > 0p>0. For a scheme X\mathfrak{X} over F_(p)\mathbb{F}_{p}, we have a Frobenius morphism Fr:XrarrX\mathrm{Fr}: \mathfrak{X} \rightarrow \mathfrak{X} induced from the pp th power map. We have a natural map Fr^(**)O_(X)rarrO_(X)\operatorname{Fr}^{*} \mathcal{O}_{\mathfrak{X}} \rightarrow \mathcal{O}_{\mathfrak{X}} that induces a map O_(X)rarrFr_(**)O_(X)\mathcal{O}_{\mathfrak{X}} \rightarrow \mathrm{Fr}_{*} \mathcal{O}_{\mathfrak{X}} by adjunction. The Frobenius splitting phi:Fr_(**)O_(X)rarrO_(X)\phi: \operatorname{Fr}_{*} \mathcal{O}_{\mathfrak{X}} \rightarrow \mathcal{O}_{\mathfrak{X}} is an O_(X"-module map such that the composition ")\mathcal{O}_{\mathfrak{X} \text {-module map such that the composition }}
is the identity. If X\mathfrak{X} is projective (and is of finite type) and O_(X)\mathcal{O}_{\mathfrak{X}} admits a Frobenius splitting, then X\mathfrak{X} is reduced and an ample line bundle has the higher cohomology vanishing [68].
For generality on Frobenius splittings, as well as their applications to B\mathscr{B} and B_(C)^("thin ")\mathscr{B}_{C}^{\text {thin }}, we refer to Brion-Kumar [12] (note that [12] has a finite type assumption, that we drop in case the proof does not require it. In the paragraph above, reducedness does not require the finite type assumption, while the higher cohomology vanishing requires the finite type assumption through the Serre vanishing). Frobenius splitting of B\mathscr{B} in char k=p\mathbb{k}=p is useful in proving that Schubert and Richardson varieties are reduced, normal, and have rational singularities. There are two major ways to construct a Frobenius splitting of B\mathscr{B} : one is to investigate the global section of the (1-p)(1-p) th power of the canonical bundle, and the other is to use a Bott-Samelson-Demazure-Hansen (=BSDH) resolution of B\mathscr{B}.
Since B_(C)^("thin ")\mathscr{B}_{C}^{\text {thin }} is no longer smooth, we cannot use the canonical bundle to construct a Frobenius splitting. Nevertheless, a (partial) BSDH resolution does the job. The situation of B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }} is a bit worse. The canonical bundle of B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }} makes some sense, but the author does not know whether it has enough power to produce a Frobenius splitting. The scheme B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}
admits a BSDH resolution, but it is a successive P^(1)\mathbb{P}^{1}-fibration over an infinite-type scheme. Thus, we cannot equip B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }} with a Frobenius splitting by either of the above means at present. Despite this, we can transfer a Frobenius splitting of B_(C)^("thin ")\mathscr{B}_{C}^{\text {thin }} to B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }} by using the compatible splitting property of a point [49], following an idea of Mathieu.
Frobenius splitting of Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }} (or rather each of its ind-piece Q_(G)(w)\mathbf{Q}_{G}(w) ) is used below, and hence we need a recipe to produce one. However, the situation of the BSDH resolution is similar to that of B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}, and the canonical bundle on Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }} simply does not make sense naively (e.g., its TT-weight at a point must be infinity). Therefore, we need a new proof strategy. Our strategy in [50] is to regard R_(G)R_{G} as a subalgebra of the corresponding coordinate ring of B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}, and prove that a Frobenius splitting of B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }} preserves R_(G)R_{G}. For this, we first see that each W(m lambda)(m inZ_( > 0),lambda inX_(+))\mathbb{W}(m \lambda)\left(m \in \mathbb{Z}_{>0}, \lambda \in \mathbb{X}_{+}\right)is a quotient of L(m Lambda)L(m \Lambda) for some Lambda inP_(+)\Lambda \in P_{+}by twisting the G[z^(-1)]G\left[z^{-1}\right]-action into a G[z]G[z]-action as z^(-1)|->zz^{-1} \mapsto z. Let pi_(m):L(m Lambda)rarrW(m lambda)\pi_{m}: L(m \Lambda) \rightarrow \mathbb{W}(m \lambda) be the quotient map. This embeds (a suitable Z\mathbb{Z}-graded subalgebra of) R_(G)R_{G} into (3.1) as an algebra with G[[z]]|><G_(m)^("rot ")G \llbracket z \rrbracket \ltimes \mathbb{G}_{m}^{\text {rot }}-action. We need to show that the map phi^(vv)\phi^{\vee} obtained by dualizing the Frobenius splitting of B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }} induces a map phi_(W)^(vv)\phi_{\mathbb{W}}^{\vee} in the following diagram:
This is equivalent to seeing that phi^(vv)(ker pi_(m))sub ker pi_(pm)\phi^{\vee}\left(\operatorname{ker} \pi_{m}\right) \subset \operatorname{ker} \pi_{p m}. We use the projectivity of W(m lambda)\mathbb{W}(m \lambda) in ⨀(m lambda)\bigodot(m \lambda) to assume that the G[z]G[z]-module generators of ker pi_(m)\pi_{m} have TT-weights that do not appear in W(m lambda)\mathbb{W}(m \lambda). In view of the fact that ker pi_(pm)\pi_{p m} contains all the TT-weight spaces in L(pm Lambda)L(p m \Lambda) whose TT-weights do not appear in W(pm lambda)\mathbb{W}(p m \lambda), we have necessarily phi^(vv)(ker pi_(m))sub\phi^{\vee}\left(\operatorname{ker} \pi_{m}\right) \subsetker pi_(pm)\operatorname{ker} \pi_{p m} by the TT-weight comparison of the generators.
In fact, every L(Lambda)L(\Lambda) admits a filtration by global Weyl modules when char k=0\mathbb{k}=0 if we twist the action of G[z]G[z] on global Weyl modules into G[z^(-1)][51]G\left[z^{-1}\right][51]. Therefore, we indeed obtain a Frobenius splitting of Q_(G)\mathbf{Q}_{G} via a novel proof based on the "universality" of the global Weyl module W(lambda)\mathbb{W}(\lambda) explained in Section 4. In conclusion, we have:
Theorem 6.1 ([50, THeORem B]). The ind-scheme Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }} admits a Frobenius splitting that is compatible with all I\mathbf{I}-orbits when char k > 2\mathbb{k}>2.
7. CONNECTION TO THE SPACE OF RATIONAL MAPS
Keep the setting as in Section 5. Let us consider the vector space embedding k((z))subk[[z,z^(-1)]]\mathbb{k}((z)) \subset \mathbb{k} \llbracket z, z^{-1} \rrbracket into the formal power series with unbounded powers. The space k[[z,z^(-1)]]\mathbb{k} \llbracket z, z^{-1} \rrbracket no longer forms a ring. Nevertheless, we have an automorphism of k[[z,z^(-1)]]\mathbb{k} \llbracket z, z^{-1} \rrbracket by swapping zz with z^(-1)z^{-1}. Together with the Chevalley involution of GG (an automorphism of GG that sends each element of TT to its inverse), it induces an involution theta\theta on the ambient space
We remark that theta\theta induces an automorphism of GG such that B nn theta(B)=TB \cap \theta(B)=T. Let w_(0)w_{0} be the longest element in WW.
Theorem 7.1 ([50, THEOREM B]). For all w,v inW_(af)w, v \in W_{\mathrm{af}}, the scheme-theoretic intersection Q_(G)(w)nn theta(Q_(G)(vw_(0)))\mathbf{Q}_{G}(w) \cap \theta\left(\mathbf{Q}_{G}\left(v w_{0}\right)\right) is reduced (we denote this intersection by Q_(G)(w,v)\mathcal{Q}_{G}(w, v) and call it a Richardson variety of Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }} below). It is normal when char k=0\mathbb{k}=0 or char k≫0\mathbb{k} \gg 0.
The scheme Q_(G)(w,v)\mathcal{Q}_{G}(w, v) is always of finite type, and the case w,v in Ww, v \in W yields a Richardson variety of B\mathscr{B}. The normality part of the proof of Theorem 7.1 goes as follows: Our Frobenius splitting of Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }} induces a Frobenius splitting of Q_(G)(w,v)\mathcal{Q}_{G}(w, v). In particular, it is reduced and weakly normal in char k > 2\mathbb{k}>2. (Here a weakly normal ring is essentially a normal ring up to topology.) Then, we lift the weak normality to characteristic zero and prove the normality of the intersection by a geometric consideration. Once we deduce the normality in characteristic zero, we can reduce it to char k≫0\mathbb{k} \gg 0 by a general result.
Let us exhibit some relevant geometric considerations here. To this end, we assume k=C\mathbb{k}=\mathbb{C} in the rest of this section. Recall that H_(2)(B,Z)~=X^(vv)H_{2}(\mathscr{B}, \mathbb{Z}) \cong \mathbb{X}^{\vee}. Let EB_(2,beta)\mathcal{E} \mathscr{B}_{2, \beta} (resp. {:B_(2,beta))\left.\mathscr{B}_{2, \beta}\right) be the space of genus-zero stable maps with two marked points to (P^(1)xxB)(\left(\mathbb{P}^{1} \times \mathscr{B}\right)( resp. B)\mathscr{B}) whose image has class (1,beta)inH_(2)(P^(1)xxB,Z)(1, \beta) \in H_{2}\left(\mathbb{P}^{1} \times \mathscr{B}, \mathbb{Z}\right) (resp. beta inH_(2)(B,Z)\beta \in H_{2}(\mathscr{B}, \mathbb{Z}) ), regarded as an algebraic variety with rational singularities [28]. We have a subvariety EB_(2,beta)^(b)\mathscr{E} \mathscr{B}_{2, \beta}^{b} such that the first marked point lands in 0inP^(1)0 \in \mathbb{P}^{1} and the second marked point lands in oo inP^(1)\infty \in \mathbb{P}^{1} through the composition
Consider the Schubert variety (a BB-orbit closure) B(w)subB\mathscr{B}(w) \subset \mathscr{B} corresponding to w in Ww \in W and the opposite Schubert variety (a theta(B)\theta(B)-orbit closure) B^(op)(v)subB\mathscr{B}^{\mathrm{op}}(v) \subset \mathscr{B} corresponding to v in Wv \in W.
Similarly, let e_(i):B_(2,beta)rarrB(i=1,2)\mathrm{e}_{i}: \mathscr{B}_{2, \beta} \rightarrow \mathscr{B}(i=1,2) be the evaluation maps. For all w,v in Ww, v \in W and beta inX^(vv)\beta \in \mathbb{X}^{\vee}, we set B_(beta)(w,v):=(e_(1)^(-1)(B(w))nne_(2)^(-1)(B^("op ")(v)))\mathscr{B}_{\beta}(w, v):=\left(e_{1}^{-1}(\mathscr{B}(w)) \cap e_{2}^{-1}\left(\mathscr{B}^{\text {op }}(v)\right)\right). Let Q^(@)_(G)(beta)\stackrel{\circ}{Q}_{G}(\beta) denote the space of maps from P^(1)\mathbb{P}^{1} to B\mathscr{B} of degree beta\beta. By adding the identity map to P^(1)\mathbb{P}^{1}, each point of Q^(@)_(G)(beta)\stackrel{\circ}{Q}_{G}(\beta) yields a map P^(1)rarr(P^(1)xxB)\mathbb{P}^{1} \rightarrow\left(\mathbb{P}^{1} \times \mathscr{B}\right) of degree (1,beta)(1, \beta). In addition, the identification of two P^(1)\mathbb{P}^{1} 's completely determines the marked points. Hence we have an inclusion Q^(@)_(G)(beta)subEB_(2,beta)^(b)\stackrel{\circ}{Q}_{G}(\beta) \subset \mathscr{E} \mathscr{B}_{2, \beta}^{b}.
Let Q_(G)(beta)(beta inX^(vv))\mathcal{Q}_{G}(\beta)\left(\beta \in \mathbb{X}^{\vee}\right) denote the space of quasimaps from P^(1)\mathbb{P}^{1} to B\mathscr{B} of degree beta\beta [22], that is, a natural compactification of Q^(@)_(G)(beta)\stackrel{\circ}{Q}_{G}(\beta) such that
where gamma <= beta\gamma \leq \beta is defined as beta-gamma insum_(i=1)^(r)Z_( >= 0)alpha_(i)^(vv)\beta-\gamma \in \sum_{i=1}^{r} \mathbb{Z}_{\geq 0} \alpha_{i}^{\vee}, and
(P^(1))^(gamma)=prod_(i=1)^(r)((P^(1))^(m_(i))//S_(m_(i)))quad" where "gamma=sum_(i=1)^(r)m_(i)alpha_(i)^(vv)\left(\mathbb{P}^{1}\right)^{\gamma}=\prod_{i=1}^{r}\left(\left(\mathbb{P}^{1}\right)^{m_{i}} / \mathbb{S}_{m_{i}}\right) \quad \text { where } \gamma=\sum_{i=1}^{r} m_{i} \alpha_{i}^{\vee}
Here (P^(1))^(gamma)\left(\mathbb{P}^{1}\right)^{\gamma} records the place where the degree of the genuine map drops in which degree components (without ordering). By adding extra P^(1)\mathbb{P}^{1} components and (compatible) maps to B\mathscr{B} to P^(1)\mathbb{P}^{1} in (f:P^(1)rarrB)inQ^(@)_(G)(beta-gamma)\left(f: \mathbb{P}^{1} \rightarrow \mathscr{B}\right) \in \stackrel{\circ}{Q}_{G}(\beta-\gamma) at the places (and total degrees) recorded by (P^(1))^(gamma)\left(\mathbb{P}^{1}\right)^{\gamma} (for each 0 <= gamma <= beta0 \leq \gamma \leq \beta ), we obtain a map of topological spaces
that is an identity on Q^(@)_(G)(beta)\stackrel{\circ}{Q}_{G}(\beta). Givental's main lemma asserts that this is a birational morphism of integral algebraic varieties.
Proposition 7.2 ( [50,$5.2])[50, \$ 5.2]). For each beta inX^(vv)\beta \in \mathbb{X}^{\vee}, we have
as schemes. In addition, pi\pi restricts to a birational morphism
pi_(beta,w,v):EB_(beta)(w,v)rarrQ_(G)(w,vt_(beta)),quad w,v in W\pi_{\beta, w, v}: \mathscr{E} \mathscr{B}_{\beta}(w, v) \rightarrow \mathbb{Q}_{G}\left(w, v t_{\beta}\right), \quad w, v \in W
In particular, we have GB_(beta)(w,v)!=O/\mathscr{G} \mathscr{B}_{\beta}(w, v) \neq \emptyset if and only if w <= (oo)/(2)vt_(beta)w \leq \frac{\infty}{2} v t_{\beta}, and its dimension is given by the distance between ww and vt_(beta)v t_{\beta} with respect to <= (oo)/(2)\leq \frac{\infty}{2}.
In other words, the Richardson varieties of Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }} are precisely the spaces of quasimaps, possibly with additional conditions imposed by the space of stable maps. According to Buch-Chaput-Mihalcea-Perrin [13], the variety EB_(beta)(w,v)\mathscr{E} \mathscr{B}_{\beta}(w, v) is irreducible and has rational singularities if it is nonempty. Hence, we find that Q_(G)(w,vt_(beta))\mathcal{Q}_{G}\left(w, v t_{\beta}\right) is irreducible in general. Proposition 7.2 and properties of the maps pi_(beta,w,v)\pi_{\beta, w, v} are used in our proof of Theorem 7.1.
Proposition 7.2 implies that Q_(G)(w,vt_(beta))\mathcal{Q}_{G}\left(w, v t_{\beta}\right) is the closure (in Q_(G)(beta)\mathcal{Q}_{G}(\beta) ) of the space of maps from P^(1)\mathbb{P}^{1} to B\mathscr{B} such that 0,oo inP^(1)0, \infty \in \mathbb{P}^{1} land in B(w)\mathscr{B}(w) and B^(@p)(v)\mathscr{B}^{\circ p}(v), respectively. By examining the natural map EB_(beta)(w,v)rarrB_(beta)(w,v)\mathscr{E} \mathscr{B}_{\beta}(w, v) \rightarrow \mathscr{B}_{\beta}(w, v) (obtained by forgetting the map to P^(1)\mathbb{P}^{1} ), we obtain:
Corollary 7.3. For all w,v in Ww, v \in W and 0!=beta inX^(vv)0 \neq \beta \in \mathbb{X}^{\vee}, we have
dim B_(beta)(w,v)=dim EB_(beta)(w,v)-1quad" if "EB_(beta)(w,v)!=O/\operatorname{dim} \mathscr{B}_{\beta}(w, v)=\operatorname{dim} \mathscr{E} \mathscr{B}_{\beta}(w, v)-1 \quad \text { if } \mathscr{E} \mathscr{B}_{\beta}(w, v) \neq \emptyset
and B_(beta)(w,v)!=O/\mathscr{B}_{\beta}(w, v) \neq \emptyset if and only if EB_(beta)(w,v)!=O/\mathcal{E} \mathscr{B}_{\beta}(w, v) \neq \emptyset. Moreover, we have
B_(beta)(w,v)!=O/quad" and "quad dim B_(beta)(w,v)=0\mathscr{B}_{\beta}(w, v) \neq \emptyset \quad \text { and } \quad \operatorname{dim} \mathscr{B}_{\beta}(w, v)=0
if and only if w <= (oo)/(2)vt_(beta)w \leq \frac{\infty}{2} v t_{\beta} are adjacent with respect to <= (oo)/(2)\leq \frac{\infty}{2}. In such a case, B_(beta)(w,v)\mathscr{B}_{\beta}(w, v) is a point.
Thanks to the dimension axiom in quantum correlators [54, (2.5)], Corollary 7.3 describes which (primary) two-point cohomological Gromov-Witten invariant of B\mathscr{B} with respect to the Schubert bases is nonzero (we can also tell its exact value). By the divisor axiom [54, $2.2.4] and the classical Chevalley formula [16], we find the Chevalley formula in quantum cohomology of B\mathscr{B} from this [29]. This clarifies the role of Q_(G)(w,vt_(beta))\mathcal{Q}_{G}\left(w, v t_{\beta}\right) in the study of quantum cohomology of B\mathscr{B} from our perspective.
Theorem 7.4 ([47]). Let beta inX^(vv)\beta \in \mathbb{X}^{\vee} and w,v in Ww, v \in W. The variety Q_(G)(w,vt_(beta))Q_{G}\left(w, v t_{\beta}\right) has rational singularities.
Theorem 7.4 is proved by Braverman-Finkelberg [9,10][9,10] for the case w=e,v=w_(0)w=e, v=w_{0} by an analysis of Zastava spaces, which does not extend to general w,vw, v. Theorem 7.4 is the most subtle technical point in [47] and its induction steps become possible by Theorem 7.1.
8. KK-THEORETIC PETERSON ISOMORPHISM
We follow the setting of the previous section with k=C\mathbb{k}=\mathbb{C}. We understand that the KK-groups appearing here contain a suitable class of line bundles supported on subvarieties equipped with some group actions, and its scalar is extended from Z\mathbb{Z} to C\mathbb{C}. Let Gr_(G):=G((z))//G[[z]]\operatorname{Gr}_{G}:=G((z)) / G \llbracket z \rrbracket be the affine Grassmannian of GG. The set of I\mathbf{I}-orbits in Gr_(G)\mathrm{Gr}_{G} is in bijection with X^(vv)\mathbb{X}^{\vee}, while the set of G[[z]]G \llbracket z \rrbracket-orbits of Gr_(G)\mathrm{Gr}_{G} is in bijection with X_( < )^(vv)subX^(vv)\mathbb{X}_{<}^{\vee} \subset \mathbb{X}^{\vee} formed by the set of antidominant coroots. For beta inX^(vv)\beta \in \mathbb{X}^{\vee}, we set Gr_(G)(beta)subGr_(G)\operatorname{Gr}_{G}(\beta) \subset \operatorname{Gr}_{G} as the corresponding I-orbit and set Gr_(G)(beta):= bar(@)_(Gr_(G))(beta)subGr_(G)\operatorname{Gr}_{G}(\beta):=\bar{\circ}_{\mathrm{Gr}_{G}}(\beta) \subset \operatorname{Gr}_{G}. We normalize so that Gr_(G)(beta)\operatorname{Gr}_{G}(\beta) is GG-stable when beta inX_( < )^(vv)\beta \in \mathbb{X}_{<}^{\vee}, and we have dim Gr_(G)(beta)=-2|beta|\operatorname{dim} \operatorname{Gr}_{G}(\beta)=-2|\beta| in such a case, where |beta|:=sum_(i=1)^(r)beta(Ï–_(i))|\beta|:=\sum_{i=1}^{r} \beta\left(\varpi_{i}\right).
We define
K_(T)(Gr_(G)):=uuu_(beta inX^(vv))K_(T)(Gr_(G)(beta))quad" and "quadK_(G)(Gr_(G)):=uuu_(beta inXvv)K_(G)(Gr_(G)(beta))K_{T}\left(\operatorname{Gr}_{G}\right):=\bigcup_{\beta \in \mathbb{X}^{\vee}} K_{T}\left(\operatorname{Gr}_{G}(\beta)\right) \quad \text { and } \quad K_{G}\left(\operatorname{Gr}_{G}\right):=\bigcup_{\beta \in \mathbb{X} \vee} K_{G}\left(\operatorname{Gr}_{G}(\beta)\right)
These spaces are equipped with the convolution product, defined by the diagram
Gr_(G)xxGr_(G)larr^(p)G((z))xxGr_(G)rarr"q"G((z))xxxx_(I)Gr_(G)rarr"" mult ""Gr_(G)\mathrm{Gr}_{G} \times \mathrm{Gr}_{G} \stackrel{p}{\leftarrow} G((z)) \times \operatorname{Gr}_{G} \xrightarrow{q} G((z)) \times \times_{\mathbf{I}} \mathrm{Gr}_{G} \xrightarrow{\text { mult }} \mathrm{Gr}_{G}
as follows: For all cycles a,b inK_(T)(Gr_(G))~=K_(I)(Gr_(G))a, b \in K_{T}\left(\mathrm{Gr}_{G}\right) \cong K_{\mathrm{I}}\left(\mathrm{Gr}_{G}\right), we find a left I\mathbf{I}-equivariant class (a,b)(a, b) on G((z))xx_(I)Gr_(G)G((z)) \times_{\mathrm{I}} \mathrm{Gr}_{G} such that
p^(**)(a⊠b)=q^(**)(a,b)p^{*}(a \boxtimes b)=q^{*}(a, b)
This yields an associative product structure on K_(T)(Gr_(G))K_{T}\left(\mathrm{Gr}_{G}\right) that contains a zero divisor. If we restrict ourselves to K_(G)(Gr_(G))K_{G}\left(\operatorname{Gr}_{G}\right), then the algebra structure given by o.^(')\odot^{\prime} becomes commutative and integrally closed. Using an isomorphism K_(T)(pt)ox_(K_(G)(pt))K_(G)(Gr_(G))~=K_(T)(Gr_(G))K_{T}(\mathrm{pt}) \otimes_{K_{G}(\mathrm{pt})} K_{G}\left(\mathrm{Gr}_{G}\right) \cong K_{T}\left(\mathrm{Gr}_{G}\right) of K_(T)(pt)K_{T}(\mathrm{pt})-modules, we find a multiplication o.\odot of K_(T)(Gr_(G))K_{T}\left(\mathrm{Gr}_{G}\right) that extends o.^(')\odot^{\prime} on K_(G)(Gr_(G))K_{G}\left(\mathrm{Gr}_{G}\right) as a K_(T)(pt)K_{T}(\mathrm{pt})-algebra. This product o.\odot coincides with a KK-theoretic analogue of the Pontrjagin product (by the calculations in [47,$2.2][47, \$ 2.2] ). In addition, we have
[O_(Gr_(G)(beta+gamma))]=[O_(Gr_(G)(beta))]o.[O_(Gr_(G)(gamma))]quad" for "quad beta,gamma inX_( < )^(vv)\left[\mathcal{O}_{\operatorname{Gr}_{G}(\beta+\gamma)}\right]=\left[\mathcal{O}_{\operatorname{Gr}_{G}(\beta)}\right] \odot\left[\mathcal{O}_{\operatorname{Gr}_{G}(\gamma)}\right] \quad \text { for } \quad \beta, \gamma \in \mathbb{X}_{<}^{\vee}
This yields a multiplicative system in K_(T)(Gr_(G))K_{T}\left(\mathrm{Gr}_{G}\right), whose localization is denoted by K_(T)(Gr_(G))_(loc)K_{T}\left(\operatorname{Gr}_{G}\right)_{\mathrm{loc}}.
The (localized) small TT-equivariant quantum KK-group of B\mathscr{B} is defined as a vector space
We denote the variable corresponding to beta inX^(vv)\beta \in \mathbb{X}^{\vee} as Q^(beta)Q^{\beta}. The quantum KK-theoretic product ***\star is a binary operation on qK_(T)(B)_(loc)q K_{T}(\mathscr{B})_{\mathrm{loc}}, defined by Givental [33] and Lee [61], whose value (a priori) belongs to a completion of qK_(T)(B)_("loc ")q K_{T}(\mathscr{B})_{\text {loc }}. It is one of the consequence of our analysis that ***\star preserves qK_(T)(B)_("loc ")q K_{T}(\mathscr{B})_{\text {loc }}. This is usually referred to as the finiteness of the quantum KK-theoretic product (for B\mathscr{B} ) in the literature [1,13], and is one of the most fundamental questions in the study of qK_(T)(B)q K_{T}(\mathscr{B}). Lam-Li-Mihalcea-Shimozono [58] conjectured that:
Theorem 8.1 ([47]). We have an isomorphism of commutative algebras
for beta,gamma inX_( < )^(vv)\beta, \gamma \in \mathbb{X}_{<}^{\vee} such that beta(Ï–_(i)) < 0\beta\left(\varpi_{i}\right)<0 for every 1 <= i <= r1 \leq i \leq r.
Note that a presentation of the ring qK_(T)(B)q K_{T}(\mathscr{B}) for G=SL(n)G=\operatorname{SL}(n) can be read-off from Givental-Lee [34], and a presentation of the ring K_(T)(Gr_(G))K_{T}\left(\mathrm{Gr}_{G}\right) is obtained in BezrukavnikovFinkelberg-Mirković [6]. However, these are not enough to yield Theorem 8.1 (for G=SL(n))G=\mathrm{SL}(n)) as the correspondence between Schubert bases is unclear.
We have an action of the nilpotent version H^("nil ")\mathscr{H}^{\text {nil }} of the double affine Hecke algebra (associated to G)G) on K_(T)(Gr_(G))K_{T}\left(\mathrm{Gr}_{G}\right), coming from Kostant-Kumar [55]. In [47], we defined the TT-equivariant KK-group K_(T)(Q_(G)^("rat "))K_{T}\left(\mathbf{Q}_{G}^{\text {rat }}\right) of Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }} based on the construction of the (T xxG_(m)^("rot "))\left(T \times \mathbb{G}_{m}^{\text {rot }}\right) equivariant KK-group of Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }} in [52]. The I\mathbf{I}-action on Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }} induces a HH^("nil ")\mathscr{H H}^{\text {nil }}-action on K_(T)(Q_(G)^("rat "))K_{T}\left(\mathbf{Q}_{G}^{\text {rat }}\right).
The object K_(T)(Q_(G)^("rat "))K_{T}\left(\mathbf{Q}_{G}^{\text {rat }}\right) needs a completion in order to admit an action of the line bundle twists by O_(Q_(G)^("rat "))(lambda)(lambda inX)\mathcal{O}_{\mathbf{Q}_{G}^{\text {rat }}}(\lambda)(\lambda \in \mathbb{X}). It reflects the fact that the right-hand side of Theorem 5.3 (i.e., a global Weyl module) is infinite-dimensional in general, and hence the effect of oxO_(Q_(G)^("rat "))(Ï–_(i))(1 <= i <= r)\otimes \mathcal{O}_{\mathbf{Q}_{G}^{\text {rat }}}\left(\varpi_{i}\right)(1 \leq i \leq r) requires infinitely many terms to describe.
Our main idea in the proof of Theorem 8.1 is to put Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }} into the picture:
Theorem 8.2 ([47, THEOREM c]). We have a commutative diagram
that respects the Schubert bases in each object. In addition, the map Psi\Psi is an embedding of representations of HH^("nil ")\mathscr{H} \mathscr{H}^{\text {nil }}, and the map Psi\Psi intertwines the tensor product with O_(Q_(G)^("rat "))(-Ï–_(i))\mathcal{O}_{\mathbf{Q}_{G}^{\text {rat }}}\left(-\varpi_{i}\right) in K_(T)(Q_(G)^("rat "))K_{T}\left(\mathbf{Q}_{G}^{\text {rat }}\right) and the quantum product of O_(B)(-Ï–_(i))\mathcal{O}_{\mathcal{B}}\left(-\varpi_{i}\right) on qK_(T)(B)_(loc)q K_{T}(\mathscr{B})_{\mathrm{loc}} for each 1 <= i <= r1 \leq i \leq r.
The completion of K_(T)(Q_(G)^("rat "))K_{T}\left(\mathbf{Q}_{G}^{\text {rat }}\right) is compatible with the standard completion of qK_(T)(B)q K_{T}(\mathscr{B}) via the map Psi\Psi. Theorem 8.2 implies that the inverse of the operation ***O_(B)(-Ï–_(i))\star \mathcal{O}_{\mathcal{B}}\left(-\varpi_{i}\right) makes sense only after the completion of qK_(T)(B)_(loc)q K_{T}(\mathscr{B})_{\mathrm{loc}}.
Since the quantum KK-theoretic correlators (see [33,61][33,61] ) satisfy neither the dimension axiom nor divisor axiom as in the theory of quantum cohomology, the proof of Theorem 8.2
must be necessarily different from Corollary 7.3. Our construction of the map Psi\Psi is based on the following two observations:
an interpretation of the ( G_(m)\mathbb{G}_{m}-equivariant) quantum KK-theoretic correlator
valued in C[T][q^(+-1)]=C[T xxG_(m)]\mathbb{C}[T]\left[q^{ \pm 1}\right]=\mathbb{C}\left[T \times \mathbb{G}_{m}\right], for each w in W,beta inX^(vv),lambda inX_(+)w \in W, \beta \in \mathbb{X}^{\vee}, \lambda \in \mathbb{X}_{+};
for each w in W,lambda inX_(+)w \in W, \lambda \in \mathbb{X}_{+}as an element of K_(T)(Q_(G)^("rat "))K_{T}\left(\mathbf{Q}_{G}^{\text {rat }}\right).
Here we can further interpret chi(E_(B_(beta))(w,w_(0)),pi_(beta,w,w_(0))^(**)O_(Q(w,w_(0)t_(beta)))(lambda))\chi\left(\mathscr{E}_{\mathscr{B}_{\beta}}\left(w, w_{0}\right), \pi_{\beta, w, w_{0}}^{*} \mathcal{O}_{\mathscr{Q}\left(w, w_{0} t_{\beta}\right)}(\lambda)\right) using the shift operators of line bundles in quantum KK-theory [35, PROPOSITION 2.13], and hence we obtain an (abstract) presentation of qK_(T)(B)q K_{T}(\mathscr{B}) from (8.1) by the reconstruction theorem [35, PROPOSITION 2.12]. The identity (8.1) is a consequence of Theorem 7.4, and (8.2) is a consequence of compatible Frobenius splitting properties of Q_(G)(w,v)s\mathcal{Q}_{G}(w, v) \mathrm{s} and Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }} in char k > 2\mathbb{k}>2 (see the explanation about the proof of Theorem 7.1).
There is a noncommutative version of Theorem 8.2, meaning that we include G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }} (the variable " qq " above) in each item [49].
9. FUNCTORIALITY OF QUANTUM KK-GROUPS
We continue to work in the setting as in the previous section. In [50], we have presented analogues of Theorems 5.2, 5.3, and 7.1 for partial flag manifolds of GG. Let us find a standard parabolic subgroup B sub P sub GB \subset P \subset G and consider B_(P):=G//P\mathscr{B}_{P}:=G / P. Our parabolic version of the semi-infinite flag manifold Q_(G,P)^("rat ")\mathbf{Q}_{G, P}^{\text {rat }} has its set of k\mathbb{k}-valued points G((z))//(T*[P,P]((z)))G((z)) /(T \cdot[P, P]((z))). The fiber of the natural map
is isomorphic to the semi-infinite flag manifold of [L,L][L, L], where L sub PL \subset P is the maximal semisimple subgroup of PP that contains TT (the standard Levi subgroup). We also have the higher cohomology vanishing of equivariant line bundles on Q_(G,P)^("rat ")\mathbf{Q}_{G, P}^{\text {rat }} (or rather {:pi_(P)(Q_(G)))\left.\pi_{P}\left(\mathbf{Q}_{G}\right)\right) as in Theorem 5.3. These are enough to yield a morphism
that intertwines appropriate line bundle twists (and analogous quantum multiplications). This yields a diagram
where we set Q_(G,P):=pi_(P)(Q_(G))\mathbf{Q}_{G, P}:=\pi_{P}\left(\mathbf{Q}_{G}\right).
The resulting map qK_(T)(B)rarr qK_(T)(B_(P))q K_{T}(\mathscr{B}) \rightarrow q K_{T}\left(\mathscr{B}_{P}\right) is, in fact, an algebra map [48], and is easy to describe. Note that we cannot have an analogous map between ordinary KK-groups because of the higher direct images. It turns out this map sends Q^(alpha_(i)^(vv))Q^{\alpha_{i}^{\vee}} to 1 for a simple coroot alpha_(i)^(vv)\alpha_{i}^{\vee} belonging to LL, and hence is not compatible with a naive generalization of the corresponding map in the Peterson isomorphism in homology [59].
We also have a restriction map qK_(T)(B)rarr qK_(T)(B^(L))q K_{T}(\mathscr{B}) \rightarrow q K_{T}\left(\mathcal{B}^{L}\right), where B^(L):=L//(L nn B)\mathcal{B}^{L}:=L /(L \cap B) is the flag manifold of a standard Levi subgroup. This map extends to algebra maps [45]
Compared with the theory of flag manifolds, many precise results and constructions for Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }} are still missing. The most accessible set of problems might be to spell out analogues of numerous explicit formulas in classical Schubert calculus purely combinatorially by admitting geometric conclusions from [3,45,47,48,52][3,45,47,48,52] partly explained in the previous two sections. We close this note by briefly discussing some of other problems.
10.1. Categorifications of the coordinate rings
The homogeneous coordinate rings of Schubert varieties of a usual flag manifold, that are BB-stable quotient rings of (2.1), can be seen as the Grothendieck groups of suitable categories equipped with cluster structures ([60]; see also Section 3.1). Hence, it is natural to expect categorifications of the homogeneous coordinate rings of Q_(G)^("rat ")(w)\mathbf{Q}_{G}^{\text {rat }}(w) and B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}. See also [21] and [43] for related problems and partial answers.
10.2. Peterson isomorphism in quantum cohomology
The Peterson isomorphism in quantum cohomology [59,74] is an analogue of Theorem 8.1 for homology. We may apply Corollary 7.3 to [69] (that is an essential ingredient in [59]) to utilize Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }} in its proof (that looks similar to the original strategy in [74]). However, we do not know an analogue of Theorem 8.2 as we lack a proper definition of H^(∙)(Q_(G))H^{\bullet}\left(\mathbf{Q}_{G}\right).
10.3. Constructible sheaves on semi-infinite flags
In representation-theoretic analysis on B\mathscr{B}, we sometimes encounter constructible sheaves that are not NN-equivariant. Also, we want some notion of (co)homology of Q_(G)\mathbf{Q}_{G} in
Section 10.2. Therefore, it is desirable to understand constructible sheaves on Q_(G)\mathbf{Q}_{G} following [7]. The resulting objects should have connection to [30]. Note that the combinatorics that should be satisfied by the I-equivariant sheaves (equipped with Frobenius endomorphisms) have been worked out in detail [62,65][62,65].
10.4. Tensor product decompositions
The tensor product decomposition of rational representations of GG is deeply connected with our whole story due to the presentation (2.1). In [57], the geometry of flag varieties is used to deduce subtle information on the tensor products beyond the classical Littlewood-Richardson rule. It would be interesting to pursue their analogues in Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}, possibly utilizing some modular interpretation [11] and connecting with the perspectives in [5].
10.5. The cotangent bundle of semi-infinite flags
A version of the cotangent bundle of Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }} would make it possible to compare our results with the perspectives in [21,67,73]. In addition, its quantization should realize some numerics in Section 10.3. The author hopes to say a bit more on this in St. Petersburg.
ACKNOWLEDGMENTS
The works presented here could not be carried out without suggestions and interest by Misha Finkelberg. The author would like to express his deepest gratitude to him. The author also would like to thank Ivan Cherednik and Thomas Lam for sharing their insights over years, and Noriyuki Abe and Toshiyuki Tanisaki for their comments.
FUNDING
This work was partially supported by JSPS KAKENHI Grant Numbers JP26287004, JP19H01782, and JPJSBP120213210.
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SYU KATO
Department of Mathematics, Kyoto University, Oiwake Kita-Shirakawa, Sakyo, Kyoto 6068502, Japan, syuchan@math.kyoto-u.ac.jp
CHARACTER ESTIMATES FOR FINITE SIMPLE GROUPS AND APPLICATIONS
MICHAEL J. LARSEN
ABSTRACT
Let GG be a finite simple group, chi\chi an irreducible complex character, and gg an element of GG. It is often desirable to have upper bounds for |chi(g)||\chi(g)| in terms of chi(1)\chi(1) and some measure of the regularity of gg. This paper reviews what is known in this direction and presents typical applications of such bounds: to proving certain products of conjugacy classes cover GG, to solving word equations over GG, and to counting homomorphisms from a Fuchsian group to GG.
Let GG be a finite group, chi\chi the character of an irreducible complex representation rho\rho of GG, and gg an element of GG. As the eigenvalues of rho(g)\rho(g) are roots of unity, the bound |chi(g)| <= chi(1)|\chi(g)| \leq \chi(1) is trivial. For central elements gg, no stronger upper bound than chi(1)\chi(1) is possible. However, according to Schur, we know that
sum_(g in G)chi(g) bar(chi(g))=|G|\sum_{g \in G} \chi(g) \overline{\chi(g)}=|G|
and since chi(x)=chi(g)\chi(x)=\chi(g) for all xx in the conjugacy class g^(G)g^{G}, we obtain the centralizer bound
Other known upper bounds typically hold only for special classes of groups.
This paper reviews what is known about character bounds when GG is a finite simple group or is closely related to such a group. There is a substantial literature on upper bounds for character ratios (|chi(g)|)/(chi(1))\frac{|\chi(g)|}{\chi(1)}; see Martin Liebeck's survey [29] for recent results and applications in the case of groups of Lie type. These bounds are typically weakest for characters chi\chi of low degree, which points to the desirability of exponential bounds, that is, bounds of the form |chi(g)| <= chi(1)^(alpha(g))|\chi(g)| \leq \chi(1)^{\alpha(g)}, where the size of alpha(g)\alpha(g) is typically related to the size of the centralizer of gg compared to |G||G|. The next two sections focus on alternating groups and groups of Lie type, respectively. The remaining sections give some applications of these results and present some open problems.
2. SYMMETRIC AND ALTERNATING GROUPS
Motivated by questions in probability theory, a number of people have considered character ratio bounds for symmetric groups. In this series of groups, unlike groups of Lie type, character ratios for nontrivial elements and nontrivial characters can be arbitrarily close to 1 . The worst case for G=S_(n)G=\mathrm{S}_{n} is the ratio (n-3)/(n-1)\frac{n-3}{n-1}, achieved when gg is a transposition and chi\chi is a character of degree n-1n-1. Persi Diaconis and Mehrdad Shahshahani considered the case that gg is a transposition and chi\chi is any irreducible character, proving in [4] that if both the first row and the first column of the Young diagram for chi=chi_(lambda)\chi=\chi_{\lambda} have length <= n//2\leq n / 2, then the character ratio is less than 1//21 / 2, while if, for instance, the first row satisfies lambda_(1) > n//2\lambda_{1}>n / 2, then
A similar bound was given by Leopold Flatto, Andrew Odlyzko, and David Wales [8, Ñ‚HEoREM 5.2].
Yuval Roichman [39] gave a character bound of the form
(|chi(g)|)/(chi(1)) <= max(lambda_(1)//n,lambda_(1)^(')//n,c)^(supp(g))\frac{|\chi(g)|}{\chi(1)} \leq \max \left(\lambda_{1} / n, \lambda_{1}^{\prime} / n, c\right)^{\operatorname{supp}(g)}
where supp(g)\operatorname{supp}(g) denotes the number of elements of {1,dots,n}\{1, \ldots, n\} not fixed by gg, and c < 1c<1 is an absolute constant. This reflects the fact that elements with high support tend to have
small centralizers. The bound is quite good when chi\chi has small degree. However, for large nn, most characters of S_(n)\mathrm{S}_{n} have degree greater than A^(n)A^{n} for any fixed AA, and for such characters, Roichman's bound is weaker than the centralizer bound for most elements g in Gg \in G.
Philippe Biane [3] gave character ratio bounds for elements of bounded support and "balanced" characters, namely those where lambda_(1)//sqrtn\lambda_{1} / \sqrt{n} and lambda_(1)^(')//sqrtn\lambda_{1}^{\prime} / \sqrt{n} are bounded. By the work of Logan-Shepp [34] and Veršik-Kerov [44], high degree characters are typically balanced. To be more precise, this is true for characters chosen randomly, weighted by the Plancherel measure. Amarpreet Rattan and Piotr Śniady [38] generalized Biane's character bound so it applies whenever supp(g)\operatorname{supp}(g) is small enough compared to nn; if gg cannot be expressed as the product of less than pi\pi transpositions, then
which simultaneously improves on the results of [39] and [38].
Thomas Müller and Jan-Christoph Schlage-Puchta gave a character bound of exponential type [37, THEOREM B] which is good in a wide variety of situations. They proved that |chi(g)| <= chi(1)^(alpha(g))|\chi(g)| \leq \chi(1)^{\alpha(g)}, where
Being exponential, it works well whether chi(1)\chi(1) is large or small. The exponent is optimal, up to a multiplicative constant, for elements gg consisting of many cycles, for instance, for involutions. However, it can be greatly improved upon for elements consisting of few cycles. In particular, alpha(g)\alpha(g) is no smaller when gg is an nn-cycle than when it is of shape 2^(n//2)2^{n / 2}.
Sergey Fomin and Nathan Lulov [9] gave a bound specifically for elements gg of the shape r^(n//r)r^{n / r}. For fixed rr and varying nn, it takes the form
so it is essentially a bound of exponential type. Aner Shalev and I gave an exponential bound [22] for elements gg of arbitrary shape 1^(a_(1))2^(a_(2))dots1^{a_{1}} 2^{a_{2}} \ldots which is roughly comparable in strength to the Fomin-Lulov bound. Define the sequence e_(1),e_(2),dotse_{1}, e_{2}, \ldots such that for all k >= 1k \geq 1,
n^(e_(1)+cdots+e_(k))=sum_(i=1)^(k)ia_(i)n^{e_{1}+\cdots+e_{k}}=\sum_{i=1}^{k} i a_{i}
This result is stronger than the exponential bound of Müller-Schlage-Puchta for almost all elements but inferior to it when the number of fixed points of gg is very large.
None of these bounds can compete with the centralizer bound for elements consisting of very few cycles, for instance, for nn-cycles, where the centralizer bound gives |chi(g)| <= sqrtn|\chi(g)| \leq \sqrt{n}. For such elements, the Murnaghan-Nakayama rule asserts |chi(g)| <= 1|\chi(g)| \leq 1, which is obviously optimal.
From symmetric group bounds, we easily obtain alternating group bounds of comparable strength. Recall that for lambda!=lambda^(')\lambda \neq \lambda^{\prime}, the characters chi_(lambda)\chi_{\lambda} and chi_(lambda^('))\chi_{\lambda^{\prime}} restrict to the same irreducible character of A_(n)\mathrm{A}_{n}. All other irreducible characters of A_(n)\mathrm{A}_{n} arise from partitions satisfying lambda=lambda^(')\lambda=\lambda^{\prime}; for each such lambda\lambda, the restriction of chi_(lambda)\chi_{\lambda} to A_(n)\mathrm{A}_{n} decomposes as a sum of two irreducibles chi_(lambda)^(1)\chi_{\lambda}^{1} and chi_(lambda)^(2)\chi_{\lambda}^{2}. The chi_(lambda)^(i)\chi_{\lambda}^{i} take the character value chi_(lambda)(g)//2\chi_{\lambda}(g) / 2 for all g inS_(n)\\Cg \in \mathrm{S}_{n} \backslash C, where CC is a single S_(n)\mathrm{S}_{n}-conjugacy class which decomposes into two A_(n)\mathrm{A}_{n}-conjugacy classes. For elements of CC, a theorem of Frobenius gives character values, which are of the form
where n_(i)=lambda_(i)-in_{i}=\lambda_{i}-i for 1 <= i <= k1 \leq i \leq k. Character degree estimates, like those in [22], now imply that |chi_(lambda)^(i)(g)| <= chi_(lambda)(1)^(epsi)\left|\chi_{\lambda}^{i}(g)\right| \leq \chi_{\lambda}(1)^{\varepsilon} whenever nn is sufficiently large compared to epsi > 0\varepsilon>0.
3. GROUPS OF LIE TYPE
Character estimates for finite simple groups of Lie type go back to the work of David Gluck [13-15]. Unlike in the case of alternating and symmetric groups, there is a uniform bound [15] on character ratios for nontrivial characters and nontrivial gg, namely
When the cardinality qq of the field of definition of GG is large, this upper bound can be improved; Gluck [14] gives an upper bound of the form C//sqrtqC / \sqrt{q} for large qq. The qq-exponent is optimal, since for odd q,PSL_(2)(q)q, \operatorname{PSL}_{2}(q) has characters of degree (q+1)/(2)\frac{q+1}{2} or (q-1)/(2)\frac{q-1}{2}, and the value of such a character at a nontrivial unipotent element gg is (+-1+-sqrt((-1)^((q-1)/(2))q))/(2)\frac{ \pm 1 \pm \sqrt{(-1)^{\frac{q-1}{2}} q}}{2}.
If GG is a finite simple group of bounded rank, then chi(1) < |G|=O(q^(D))\chi(1)<|G|=O\left(q^{D}\right), where DD denotes the dimension associated to the Lie type of GG. Therefore, the Gluck bound C//sqrtqC / \sqrt{q} can be converted to an exponential bound |chi(g)| <= chi(1)^(alpha)|\chi(g)| \leq \chi(1)^{\alpha}, where alpha < 1\alpha<1 depends only on the rank. To achieve exponential bounds in general, therefore, it suffices to limit our attention to the case that GG is a classical group, that is, one of PSL_(r+1)(q),PSU_(r+1)(q),POmega(+-)/(2r)(q)\operatorname{PSL}_{r+1}(q), \operatorname{PSU}_{r+1}(q), \mathrm{P} \Omega \frac{ \pm}{2 r}(q), PSp_(2r)(q)\mathrm{PSp}_{2 r}(q), or POmega_(2r+1)(q)\mathrm{P} \Omega_{2 r+1}(q).
We cannot expect that character ratios go to 0 as the order of a classical group goes to infinity. For instance, let G=PSL_(r+1)(q)G=\mathrm{PSL}_{r+1}(q). The permutation representation associated with the action of GG on PF_(q)^(r)\mathbb{P} F_{q}^{r} can be expressed as chi+1\chi+1, for chi\chi irreducible. Let gg be the image of a transvection in SL_(r+1)(F_(q))\operatorname{SL}_{r+1}\left(\mathbb{F}_{q}\right) in GG. Then the fixed points of gg form a hyperplane in PP_(q)^(n)\mathbb{P P}_{q}^{n}, and chi(g)=q^(n-1)+q^(n-2)+cdots+q\chi(g)=q^{n-1}+q^{n-2}+\cdots+q. Thus,
Defining the support supp(g)\operatorname{supp}(g) as the smallest codimension of any eigenspace of gg for the natural projective representation of GG, the elements gg in the above example have constant support 1 even as the rank of GG goes to infinity. Shalev, Pham Huu Tiep, and I proved [24, THEOREM 4.3.6] that as the support goes to infinity, the character ratio goes to 0 :
This falls well short of a uniform exponential character bound, even for elements of maximal support. Robert Guralnick, Tiep, and I found uniform exponential bounds for elements gg whose centralizer is small compared to the order of GG. For instance, we proved [16, THEOREM 1.4] that if GG is of the form PSL_(n)(q)\operatorname{PSL}_{n}(q) or PSU_(n)(q)\operatorname{PSU}_{n}(q) and |C_(G)(g)| <= q^(n^(2)//12)\left|C_{G}(g)\right| \leq q^{n^{2} / 12}, then |chi(g)| <= chi(1)^(8//9)|\chi(g)| \leq \chi(1)^{8 / 9}. More generally, but less explicitly, we proved [17, THEOREM 1.3] that for all epsi > 0\varepsilon>0, there exists delta > 0\delta>0 such that |C_(G)(g)| <= |G|^(delta)\left|C_{G}(g)\right| \leq|G|^{\delta} implies |chi(g)| <= chi(1)^(epsi)|\chi(g)| \leq \chi(1)^{\varepsilon}. However, the method of these papers applies only to elements with small centralizer, for instance, it does not give any bound at all for involutions.
This defect was remedied in the sequel [28], which proved that for all positive delta < 1\delta<1 there exists epsi < 1\varepsilon<1 such that |C_(G)(g)| <= |G|^(delta)\left|C_{G}(g)\right| \leq|G|^{\delta} implies |chi(g)| <= chi(1)^(epsi)|\chi(g)| \leq \chi(1)^{\varepsilon}. More precisely, |chi(g)| <=|\chi(g)| \leqchi(1)^(alpha(g))\chi(1)^{\alpha(g)} where
and c > 0c>0 is an absolute constant, which can be made explicit (but is, unfortunately, extremely small). This theorem holds more generally for quasisimple finite groups of Lie type.
For many elements gg in a classical group of rank rr, much better exponents are available, thanks to the work of Roman Bezrukavnikov, Liebeck, Shalev, and Tiep [2]. For qq odd, if the centralizer of gg is a proper split Levi subgroup, then |chi(g)| <= f(r)chi(1)^(alpha(g))|\chi(g)| \leq f(r) \chi(1)^{\alpha(g)}, where alpha(g)\alpha(g) is an explicitly computable rational number which is known to be optimal in many cases. This idea was further developed by Jay Taylor and Tiep, who proved [43], among other things, that for every nontrivial element g inPSL_(n)(q)g \in \operatorname{PSL}_{n}(q),
All of these estimates are poor for elements with small centralizers, such as regular elements. A general result, due to Shelly Garion, Alexander Lubotzky, and myself, which sometimes gives reasonably good bounds for regular elements, is the following [10, THEOREM 3]. Let GG be a finite group, not necessarily simple, and gg an element of GG whose centralizer AA is abelian. Suppose A_(1),dots,A_(n)A_{1}, \ldots, A_{n} are subgroups of AA not containing gg such that the centralizer of every element of A\\uuu_(i)A_(i)A \backslash \bigcup_{i} A_{i} is AA. Then, for every irreducible character of GG,
For example, this gives an upper bound of 2(n-1)^(2)//sqrt32(n-1)^{2} / \sqrt{3} for |chi(g)||\chi(g)| when G=PSL_(n)(q)G=\operatorname{PSL}_{n}(q) and gg is the image of an element with irreducible characteristic polynomial. It would be nice to have optimal upper bounds for |chi(g)||\chi(g)| for general regular semisimple elements gg.
4. PRODUCTS OF CONJUGACY CLASSES
If C_(1),dots,C_(n)C_{1}, \ldots, C_{n} are conjugacy classes of a finite group GG, then the number NN of nn tuples (g_(1),dots,g_(n))inC_(1)xx cdots xxC_(n)\left(g_{1}, \ldots, g_{n}\right) \in C_{1} \times \cdots \times C_{n} satisfying g_(1)g_(2)cdotsg_(n)=1g_{1} g_{2} \cdots g_{n}=1 is given by the Frobenius formula
where chi\chi ranges over all irreducible characters of GG. In conjunction with upper bounds for the |chi(C_(i))|\left|\chi\left(C_{i}\right)\right|, this can sometimes be used to prove that N!=0N \neq 0, as the contribution from chi=1\chi=1 often dominates the sum. Exponential bounds for the chi(C_(i))\chi\left(C_{i}\right) are especially convenient, since results of Liebeck and Shalev [32] give a great deal of information about when we can expect
A well-known conjecture attributed to Thompson asserts that for every finite simple group GG, there exists a conjugacy class CC such that C^(2)=GC^{2}=G. Thanks to work of Erich Ellers and Nikolai Gordeev [6], we know that this is true except for a list of possible counterexamples, all finite simple groups of Lie type with q <= 8q \leq 8. Tiep and I used our uniform exponential bounds to show that several of the infinite families on this list, in particular, the symplectic groups for all q >= 2q \geq 2, can be eliminated in sufficiently high rank [28, Ñ‚HEoREM 7.7]. It would be interesting if these results could be extended to the remaining families on the list, giving an asymptotic version of Thompson's conjecture.
Andrew Gleason and Cheng-hao Xu [18,19] proved Thompson's conjecture for alternating groups, using the conjugacy class of an nn-cycle if nn is odd or a permutation of shape 2^(1)(n-2)^(1)2^{1}(n-2)^{1} if nn is even. In [22, THEOREM 1.13], Shalev and I proved that in the limit n rarr oon \rightarrow \infty the probability that a randomly chosen g inA_(n)g \in \mathrm{A}_{n} belongs to a conjugacy class with C^(2)=A_(n)C^{2}=\mathrm{A}_{n} rapidly approaches 1 .
The analogous claim cannot be true for all finite simple groups since C^(2)=GC^{2}=G implies that C=C^(-1)C=C^{-1}, and for, e.g., PSL_(3)(q)\operatorname{PSL}_{3}(q) as q rarr ooq \rightarrow \infty, the probability that a random element is real goes to 0 . However, there are several variants of this question which do not have an obvious counterexample. As the order of GG tends to infinity, does the probability that a random real element belongs to a conjugacy class with C^(2)=GC^{2}=G approach 1? Does the probability that a random element gg belongs to a conjugacy class CC with C^(2)uu{1}=GC^{2} \cup\{1\}=G approach 1? Also, as the order of GG tends to infinity, does the probability that a random element belongs to a conjugacy class with CC^(-1)=GC C^{-1}=G approach 1?
The weaker claim that every element g in Gg \in G lies in CC^(-1)C C^{-1} for some conjugacy class (depending, perhaps, on gg ) is equivalent to the statement that every element of GG is a commutator. This was was an old conjecture of Ore and is now a theorem of Liebeck, Eamonn O'Brien, Shalev, and Tiep [30].
One can also ask about S^(2)S^{2} where SS is an arbitrary conjugation-invariant subset of GG. On naive probabilistic grounds, it might seem plausible that given epsi > 0\varepsilon>0 fixed, for GG sufficiently large, every normal subset of GG with at least epsi|G|\varepsilon|G| elements satisfies S^(2)=GS^{2}=G. However, a moment's reflection shows that, unless epsi > (1)/(2)\varepsilon>\frac{1}{2}, there is no reason to expect 1inS^(2)1 \in S^{2}.
Is it true, for GG sufficiently large, that S^(2)uu{1}=GS^{2} \cup\{1\}=G ? For alternating groups and for groups of Lie type in bounded rank, the answer is affirmative [26], but we do not know in general.
In a different direction, given a conjugacy class CC, how large must nn be so that the nnth power C^(n)C^{n} is all of GG ? More generally, given conjugacy classes C_(1),dots,C_(n)C_{1}, \ldots, C_{n} with sufficiently strong character bounds, the Frobenius formula can be used to show that each element of GG is represented as a product g_(1)cdotsg_(n)g_{1} \cdots g_{n}, with g_(i)inC_(i)g_{i} \in C_{i}, in approximately (|C_(1)|cdots|C_(n)|)/(|G|)\frac{\left|C_{1}\right| \cdots\left|C_{n}\right|}{|G|} ways. For instance, it follows from the exponential character bounds given above that there exists an absolute constant kk such that if GG is a finite simple group of Lie type and C_(1),dots,C_(n)C_{1}, \ldots, C_{n} are conjugacy classes in GG satisfying |C_(1)|cdots|C_(n)| > |G|^(k)\left|C_{1}\right| \cdots\left|C_{n}\right|>|G|^{k}, then for each g in Gg \in G,
Via Lang-Weil estimates, this further implies that if C__(1),dots,C__(n)\underline{C}_{1}, \ldots, \underline{C}_{n} are conjugacy classes of a simple algebraic group G_\underline{G}, and
dim C__(1)+cdots+dim C__(n) > k dim G_\operatorname{dim} \underline{C}_{1}+\cdots+\operatorname{dim} \underline{C}_{n}>k \operatorname{dim} \underline{G}
then the product morphism of varieties C__(1)xx cdots xxC__(n)rarrG_\underline{C}_{1} \times \cdots \times \underline{C}_{n} \rightarrow \underline{G} has the property that every fiber is of dimension dim C__(1)+cdots+dim C__(n)-dim G_\operatorname{dim} \underline{C}_{1}+\cdots+\operatorname{dim} \underline{C}_{n}-\operatorname{dim} \underline{G}.
In the special case that C_(1)=cdots=C_(n)=CC_{1}=\cdots=C_{n}=C, the question of the distribution of products g_(1)cdotsg_(n),g_(i)in Cg_{1} \cdots g_{n}, g_{i} \in C, can be expressed in terms of the mixing time of the random walk on the Cayley graph of ( G,CG, C ). A consequence of the exponential character bounds [28] is that for groups of Lie type, the mixing time of such a random walk is O(log |G|//log |C|)O(\log |G| / \log |C|). This is the same order of growth as the diameter of the Cayley graph, thus settling conjectures of Lubotzky [35, P. 179] and Shalev [42, CONJECTURE 4.3].
The situation is different for alternating groups G=A_(n)G=\mathrm{A}_{n}. For instance, if CC is the class of 3-cycles and n >= 6n \geq 6, then log |G|//log |C| < n\log |G| / \log |C|<n, and C^(|__ n//2__|)=GC^{\lfloor n / 2\rfloor}=G [5, THEOREM 9.1]. However, for any fixed kk, the probability that the product of knk n random 3 -cycles g_(i)g_{i} fixes 1 is at least the probability that each individual g_(i)g_{i} fixes 1 , which goes to e^(-3k)e^{-3 k} as n rarr oon \rightarrow \infty. Thus the expected number of fixed points of g_(1)cdotsg_(n)g_{1} \cdots g_{n} grows linearly with nn. It would be interesting to know, for general C subA_(n)C \subset \mathrm{A}_{n}, what the mixing time is.
5. WARING'S PROBLEM
Waring's problem for finite simple groups originally meant the following question. Does there exist a function f:NrarrNf: \mathbb{N} \rightarrow \mathbb{N} such that for all positive integers nn and all sufficiently large finite simple groups GG (in terms of nn ), every element of GG is a product of f(n)nf(n) nth powers? Positive solutions were given by Martinez-Zelmanov [36] and Saxl-Wilson [40].
This can be extended as follows. Let ww denote a nontrivial element in any free group F_(d)F_{d}. For every finite simple group G,wG, w determines a function G^(d)rarr GG^{d} \rightarrow G. We replace the nnth powers with word values, that is, elements of GG in the image of ww. Liebeck and Shalev proved [31] that for GG sufficiently large (in terms of ww ), every element of GG can be written as a product of a bounded number of word values (where the bound may depend on ww, just as
in the classical version of Waring's problem, the minimum number of the nnth powers needed to represent a given integer may depend on nn ).
It was therefore, perhaps, surprising when Shalev proved [41] that the Waring number for finite simple groups is uniform in ww and is, in fact, at most three. This has now been improved to the optimal bound, two [23,24]. More generally, for any two nontrivial words w_(1)w_{1} and w_(2)w_{2}, if GG is a sufficiently large finite simple group, every element of GG is a product of their word values. In fact, it is even possible [27] to choose subsets S_(1)S_{1} and S_(2)S_{2} of the sets of word values of w_(1)w_{1} and w_(2)w_{2} such that S_(1)S_(2)=GS_{1} S_{2}=G and |S_(i)|=O(|G|^(1//2)log^(1//2)|G|)\left|S_{i}\right|=O\left(|G|^{1 / 2} \log ^{1 / 2}|G|\right). The set of values of any word is a union of conjugacy classes, and the basic strategy of the proof is to try to find conjugacy classes C_(1)C_{1} and C_(2)C_{2} contained in the word values of w_(1)w_{1} and w_(2)w_{2}, respectively, such that C_(1)C_(2)=GC_{1} C_{2}=G and very few elements of GG have significantly fewer representations as such products than one would expect. Then a random choice of subsets S_(i)subC_(i)S_{i} \subset C_{i} of suitable size can almost always be slightly modified to work.
In general, the probability distribution on the word values of ww obtained by evaluation at a uniformly distributed random element of G^(d)G^{d} is far from uniform. For instance, for g inA_(3n)g \in \mathrm{A}_{3 n} uniformly distributed, the probability that g^(3)=1g^{3}=1 is at least |A_(3n)|^(-1)\left|\mathrm{A}_{3 n}\right|^{-1} times the number of elements of shape 3^(n)3^{n}, i.e.,
for nn sufficiently large. Thus, setting w_(1)=w_(2)=x^(3)w_{1}=w_{2}=x^{3}, the probability that the product of cubes of two randomly chosen elements is 1 is at least |A_(3n)|^(-2//3-2//3n)\left|A_{3 n}\right|^{-2 / 3-2 / 3 n}, which, for large nn, makes the distribution far from uniform, at least in the L^(oo)L^{\infty} sense.
Using exponential character estimates, Shalev, Tiep, and I proved [25, theorem 4] that for any word ww, there exists kk such that as |G|rarr oo|G| \rightarrow \infty, the L^(oo)L^{\infty}-deviation from uniformity in the product of kk independent randomly generated values of ww goes to 0 . The dependence of kk on ww is unavoidable, as the above example suggests. On the other hand, the L^(1)L^{1}-deviation from uniformity goes to 0 in the product of two independent randomly generated values of ww, for any nontrivial word ww [25, THEOREM 1]. I do not know what to expect for L^(p)L^{p}-deviation for 1 < p < oo1<p<\infty.
6. FUCHSIAN GROUPS
For g,m >= 0g, m \geq 0, let d_(1),dots,d_(m) >= 2d_{1}, \ldots, d_{m} \geq 2 be integers. For
Assume e < 0e<0, so Gamma\Gamma is an oriented, cocompact Fuchsian group. Let GG be a finite group, and let C_(1),dots,C_(m)C_{1}, \ldots, C_{m} denote conjugacy classes in GG of elements whose orders divide d_(1),dots,d_(m)d_{1}, \ldots, d_{m},
respectively. The Frobenius formula can be regarded as the g=0g=0 case of a more general formula for the number of homomorphisms Gamma rarr G\Gamma \rightarrow G mapping x_(i)x_{i} to an element of C_(i)C_{i} for all ii,
In favorable situations, one can prove that the chi=1\chi=1 term dominates all the others combined, in which case one has a good estimate for the number of such homomorphisms. Using this, Liebeck and Shalev proved [32, THEOREM 1.5] that if g >= 2g \geq 2, and GG is a simple of Lie type group of rank rr, then
By the same method, employing the character bounds of [28], one obtains the same estimate whenever ee is less than some absolute constant, regardless of the value of gg. It would be interesting to know whether this is true in general for e < 0e<0. Some evidence in favor of this idea is given in [21,33], but for small qq the problem is open.
An interesting geometric consequence of the method of Liebeck-Shalev is that if G_\underline{G} is a simple algebraic group of rank rr and g >= 2g \geq 2, the morphism G_^(2g)rarrG_\underline{G}^{2 g} \rightarrow \underline{G} given by the word [y_(1),z_(1)]cdots[y_(g),z_(g)]\left[y_{1}, z_{1}\right] \cdots\left[y_{g}, z_{g}\right] has all fibers of the same dimension, (2g-1)dim G_(2 g-1) \operatorname{dim} \underline{G}. This has been refined by Avraham Aizenbud and Nir Avni, who proved [1] that for g >= 373g \geq 373, the fibers of this morphism are reduced and have rational singularities. It would be interesting to extend this to the case of general Fuchsian groups. For instance, does there exist an absolute constant k\mathrm{k} such that for all simple algebraic groups G_\underline{G} and conjugacy classes C__(1),dots,C__(m)\underline{C}_{1}, \ldots, \underline{C}_{m} with dim C__(1)+\operatorname{dim} \underline{C}_{1}+cdots+dim C__(m) > k dim G_\cdots+\operatorname{dim} \underline{C}_{m}>k \operatorname{dim} \underline{G}, all fibers of the multiplication morphism C__(1)xx cdots xxC__(m)rarrG_\underline{C}_{1} \times \cdots \times \underline{C}_{m} \rightarrow \underline{G} are reduced with rational singularities The ideas of Glazer-Hendel [11,12] may be applicable.
For g=1g=1, we can no longer hope for equidimensional fibers, since the generic fiber dimension is dim G_\operatorname{dim} \underline{G}, while the fiber over the identity element has dimension r+dim G_r+\operatorname{dim} \underline{G}. However, Zhipeng Lu\mathrm{Lu} and I proved [20] that for G_=SL_(n)\underline{G}=\mathrm{SL}_{n}, all fibers over noncentral elements have dimension G_\underline{G}. It would be interesting to know whether this is true for general simple algebraic groups G_\underline{G}.
FUNDING
This work was partially supported by the National Science Foundation.
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MICHAEL J. LARSEN
Department of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A., mjlarsen@indiana.edu
FINITE APPROXIMATIONS AS A TOOL FOR STUDYING TRIANGULATED CATEGORIES
AMNON NEEMAN
Abstract
Small, finite entities are easier and simpler to manipulate than gigantic, infinite ones. Consequently, huge chunks of mathematics are devoted to methods reducing the study of big, cumbersome objects to an analysis of their finite building blocks. The manifestation of this general pattern, in the study of derived and triangulated categories, dates back almost to the beginnings of the subject-more precisely to articles by Illusie in SGA6, way back in the early 1970s.
What is new, at least new in the world of derived and triangulated categories, is that one gets extra mileage from analyzing more carefully and quantifying more precisely just how efficiently one can estimate infinite objects by finite ones. This leads one to the study of metrics on triangulated categories, and of how accurately an object can be approximated by finite objects of bounded size.
In every branch of mathematics, we try to solve complicated problems by reducing to simpler ones, and from antiquity people have used finite approximations to study infinite objects. Naturally, whenever a new field comes into being, one of the first developments is to try to understand what should be the right notion of finiteness in the discipline. Derived and triangulated categories were introduced by Verdier in his PhD thesis in the mid-1960s (although the published version only appeared much later in [38]). Not surprisingly, the idea of studying the finite objects in these categories followed suit soon after, see Illusie [13-15].
Right from the start there was a pervasive discomfort with derived and triangulated categories-the intuition that had been built up, in dealing with concrete categories, mostly fails for triangulated categories. In case the reader is wondering: in the previous sentence the word "concrete" has a precise, technical meaning, and it is an old theorem of Freyd [10,11] that triangulated categories often are not concrete. Further testimony, to the strangeness of derived and triangulated categories, is that it took two decades before the intuitive notion of finiteness, which dates back to Illusie's articles [13-15], was given its correct formal definition. The following may be found in [23, DEFINITION 1.1].
Definition 1.1. Let T\mathscr{T} be a triangulated category with coproducts. An object C inTC \in \mathscr{T} is called compact if Hom(C,-)\operatorname{Hom}(C,-) commutes with coproducts. The full subcategory of all compact objects will be denoted by T^(c)\mathscr{T}^{c}.
Remark 1.2. I have often been asked where the name "compact" came from. In the preprint version of [23], these objects went by a different name, but the (anonymous) referee did not like it. I was given a choice: I was allowed to baptize them either "compact" or "small."
Who was I to argue with a referee?
Once one has a good working definition of what the finite objects ought to be, the next step is to give the right criterion which guarantees that the category has "enough" of them. For triangulated categories, the right definition did not come until [24, DEFINITION 1.7].
Definition 1.3. Let T\mathscr{T} be a triangulated category with coproducts. The category T\mathscr{T} is called compactly generated if every nonzero object X inTX \in \mathscr{T} admits a nonzero map C rarr XC \rightarrow X, with C inTC \in \mathscr{T} a compact object.
As the reader may have guessed from the name, compactly generated triangulated categories are those in which it is often possible to reduce general problems to questions about compact objects-which tend to be easier.
All of the above nowadays counts as "classical," meaning that it is two or more decades old and there is already a substantial and diverse literature exploiting the ideas. This article explores the recent developments that arose from trying to understand how efficiently one can approximate arbitrary objects by compact ones. We first survey the results obtained to date. This review is on the skimpy side, partly because there already are other, more expansive published accounts in the literature, but mostly because we want to leave ourselves space to suggest possible directions for future research. Thus the article can be
thought of as having two components: a bare-bone review of what has been achieved to date, occupying Sections 2 to 6 , followed by Section 7 which comprises suggestions of avenues that might merit further development.
Our review presents just enough detail so that the open questions, making up Section 7, can be formulated clearly and comprehensibly, and so that the significance and potential applications of the open questions can be illuminated. This has the unfortunate side effect that we give short shrift to the many deep, substantial contributions, made by numerous mathematicians, which preceded and inspired the work presented here. The author apologizes in advance for this omission, which is the inescapable corollary of page limits. The reader is referred to the other surveys of the subject, where more care is taken to attribute the ideas correctly to their originators, and give credit where credit is due.
We permit ourselves to gloss over difficult technicalities, nonchalantly skating by nuances and subtleties, with only an occasional passing reference to the other surveys or to the research papers for more detail.
The reader wishing to begin with examples and applications, to keep in mind through the forthcoming abstraction, is encouraged to first look at the Introduction to [31].
2. APPROXIMABLE TRIANGULATED CATEGORIES-THE FORMAL DEFINITION AS A VARIANT ON FOURIER SERIES
It is now time to start our review, offering a glimpse of the recent progress that was made by trying to measure how "complicated" an object is, in other words, how far it is from being compact. What follows is sufficiently new for there to be much room for improvement: the future will undoubtedly see cleaner, more elegant, and more general formulations. What is presented here is the current crude state of this emerging field.
Discussion 2.1. This section is devoted to defining approximable triangulated categories, and the definition is technical and at first sight could appear artificial, maybe even forbidding, It might help therefore to motivate it with an analogy.
Let S^(1)\mathbb{S}^{1} be the circle, and let M(S^(1))M\left(\mathbb{S}^{1}\right) be the set of all complex-valued, Lebesguemeasurable functions on S^(1)\mathbb{S}^{1}. As usual we view S^(1)=R//Z\mathbb{S}^{1}=\mathbb{R} / \mathbb{Z} as the quotient of its universal cover R\mathbb{R} by the fundamental group Z\mathbb{Z}; this identifies functions on S^(1)\mathbb{S}^{1} with periodic functions on R\mathbb{R} with period 1. In particular the function g(x)=e^(2pi ix)g(x)=e^{2 \pi i x} belongs to M(S^(1))M\left(\mathbb{S}^{1}\right). And, for each ℓinZ\ell \in \mathbb{Z}, we have that g(x)^(ℓ)=e^(2pi iℓx)g(x)^{\ell}=e^{2 \pi i \ell x} also belongs to M(S^(1))M\left(\mathbb{S}^{1}\right). Given a norm on the space M(S^(1))M\left(\mathbb{S}^{1}\right), for example, the L^(p)L^{p}-norm, we can try to approximate arbitrary f in M(S^(1))f \in M\left(\mathbb{S}^{1}\right) by Laurent polynomials in gg, that is, look for complex numbers {lambda_(ℓ)inC∣-n <= ℓ <= n}\left\{\lambda_{\ell} \in \mathbb{C} \mid-n \leq \ell \leq n\right\} such that
with epsi > 0\varepsilon>0 small. This leads us to the familiar territory of Fourier series.
Now imagine trying to do the same, but replacing M(S^(1))M\left(\mathbb{S}^{1}\right) by a triangulated category. Given a triangulated category T\mathscr{T}, which we assume to have coproducts, we would like to pretend to do Fourier analysis on it. We would need to choose:
(1) Some analog of the function g(x)=e^(2pi ix)g(x)=e^{2 \pi i x}. Our replacement for this will be to choose a compact generator G inTG \in \mathscr{T}. Recall that a compact generator is a compact object G inTG \in \mathscr{T} such that every nonzero object X inTX \in \mathscr{T} admits a nonzero map G[i]rarr XG[i] \rightarrow X for some i inZi \in \mathbb{Z}.
(2) We need to choose something like a metric, the analog of the L^(p)L^{p}-norm on M(S^(1))M\left(\mathbb{S}^{1}\right). For us this will be done by picking a t-structure (T <= 0,T^( >= 0))\left(\mathscr{T} \leq 0, \mathscr{T}^{\geq 0}\right) on T\mathscr{T}. The heuristic is that we will view a morphism E rarr FE \rightarrow F in T\mathscr{T} as "short" if, in the triangle E rarr F rarr DE \rightarrow F \rightarrow D, the object DD belongs to T^( <= -n)\mathscr{T}^{\leq-n} for large nn. We will come back to this in Discussion 6.10.
(3) We need to have an analog of the construction that passes, from the function g(x)=e^(2pi ix)g(x)=e^{2 \pi i x} and the integer n > 0n>0, to the vector space of trigonometric Laurent polynomials sum_(â„“=-n)^(n)lambda_(â„“)e^(2pi iâ„“x)\sum_{\ell=-n}^{n} \lambda_{\ell} e^{2 \pi i \ell x}.
As it happens our solution to (3) is technical. We need a recipe that begins with the object GG and the integer n > 0n>0, and proceeds to cook up a collection of more objects. We ask the reader to accept it as a black box, with only a sketchy explanation just before Remark 2.3.
Black Box 2.2. Let T\mathscr{T} be a triangulated category and let G inTG \in \mathscr{T} be an object. Let n > 0n>0 be an integer. We will have occasion to refer to the following four full subcategories of T\mathscr{T} :
(1) The subcategory (:G:)_(n)subT\langle G\rangle_{n} \subset \mathscr{T} is defined unconditionally, and if T\mathscr{T} has coproducts one can also define the larger subcategory bar((:G:))_(n)\overline{\langle G\rangle}_{n}. Both of these subcategories are classical, the reader can find the subcategory (:G:)_(n)\langle G\rangle_{n} in Bondal and Van den Bergh [6, THE DISCUSSION BETWEEN LEMMA 2.2.2 AND DEFINITION 2.2.3], and the subcategory bar((:G:))\overline{\langle G\rangle} in [6, THE DISCUSSION BETWEEN DEFINITION 2.2.3 AND PROPOSITION 2.2.4].
(2) If the category T\mathscr{T} has coproducts, we will also have occasion to consider the full subcategory bar((:G:))(-oo,n]\overline{\langle G\rangle}(-\infty, n]. Once again this category is classical (although the name is not). The reader can find it in Alonso, JeremÃas, and Souto [1], where it would go by the name "the cocomplete pre-aisle generated by G[-n]G[-n] ".
(3) Once again assume that T\mathscr{T} has coproducts. Then we will also look at the full
Below we give a vague description of what is going on in these constructions; but when it comes to the technicalities, we ask the reader to either accept these as black boxes, or refer to [29, REMINDER 0.8 (VII), (XI) AND (XII)] for detail. We mention that there is a slight clash of notation in the literature: what we call bar((:G:))\overline{\langle G\rangle} in (1), following Bondal and Van den Bergh,
goes by a different name in [29, REMINDER 0.8 (XI)]. The name it goes by there is the case A=-ooA=-\infty and B=ooB=\infty of the more general subcategory bar((:G:))_(n)^([A,B])\overline{\langle G\rangle}_{n}^{[A, B]}.
Now for the vague explanation of what goes on in (1), (2), and (3) above: in a triangulated category T\mathscr{T}, there are not many ways to build new objects out of old ones. One can shift objects, form direct summands, form finite direct sums (or infinite ones if coproducts exist), and one can form extensions. In the categories (:G:)_(n)\langle G\rangle_{n} and bar((:G:)_(n))\overline{\langle G\rangle_{n}} of (1), there is a bound on the number of allowed extensions, and the difference between the two is whether infinite coproducts are allowed. In the category bar((:G:))^((-oo,n])\overline{\langle G\rangle}{ }^{(-\infty, n]} of (2), the bound is on the permitted shifts. And in the category bar((:G:))_(n)^([-n,n])\overline{\langle G\rangle}_{n}^{[-n, n]} of (3), both the shifts allowed and the number of extensions permitted are restricted.
Remark 2.3. The reader should note that an example would not be illuminating, the categories (:G:)_(n), bar((:G:))_(n), bar((:G:))^((-oo,n])\langle G\rangle_{n}, \overline{\langle G\rangle}_{n}, \overline{\langle G\rangle}^{(-\infty, n]}, and bar((:G:)_(n)^([-n,n]))\overline{\langle G\rangle_{n}^{[-n, n]}} are not usually overly computable. For example, let RR be an associative ring, and let T=D(R)\mathscr{T}=\mathbf{D}(R) be the unbounded derived category of complexes of left RR-modules. The object R inTR \in \mathscr{T}, that is, the complex which is RR in degree zero and vanishes in all other degrees, is a compact generator for T=D(R)\mathscr{T}=\mathbf{D}(R).
But if we wonder what the categories (:R:)_(n), bar((:R:))_(n), bar((:R:))^((-oo,n])\langle R\rangle_{n}, \overline{\langle R\rangle}_{n}, \overline{\langle R\rangle}^{(-\infty, n]}, and bar((:R:))_(n)^([-n,n])\overline{\langle R\rangle}_{n}^{[-n, n]} might turn out to be, only the category bar((:R:))^((-oo,n])\overline{\langle R\rangle}^{(-\infty, n]} is straightforward: it is the category of all cochain complexes whose cohomology vanishes in degrees > n>n. The three categories (:R:)_(n), bar((:R:)_(n))\langle R\rangle_{n}, \overline{\langle R\rangle_{n}}, and bar((:R:))_(n)^([-n,n])\overline{\langle R\rangle}_{n}^{[-n, n]} are mysterious in general. In fact, the computation of (:G:)_(n)\langle G\rangle_{n} is the subject of conjectures that have attracted much interest. We will say a tiny bit about theorems in this direction in Section 4, and will mention one of the active, open conjectures in the discussion between Definition 7.7 and Problem 7.8.
Remark 2.4. In the definition of approximable triangulated categories, which is about to come, the category bar((:G:)_(n))[-n,n]\overline{\langle G\rangle_{n}}[-n, n] will play the role of the replacement for the vector space of trigonometric Laurent polynomials of degree <= n\leq n, which came up in the desiderata of Discussion 2.1(3). The older categories (:G:)_(n), bar((:G:))_(n)\langle G\rangle_{n}, \overline{\langle G\rangle}_{n}, and bar((:G:))(-oo,n]\overline{\langle G\rangle}(-\infty, n] will be needed later in the article.
Remark 2.5. Let us return to the heuristics of Discussion 2.1. Assume we have chosen the t-structure (T <= 0,T^(0))\left(\mathscr{T} \leq 0, \mathscr{T}^{0}\right) as in Discussion 2.1(2), which we think of as our replacement for the L^(p)L^{p}-norm on M(S^(1))M\left(\mathbb{S}^{1}\right). And we have also chosen a compact generator G inTG \in \mathscr{T} as in Discussion 2.1(1), which we think of as the analog of the exponential function g(x)=e^(2pi ix)g(x)=e^{2 \pi i x}. We have declared that the subcategories bar((:G:)_(n)^([-n,n]))\overline{\langle G\rangle_{n}^{[-n, n]}} will be our replacement for the vector space of trigonometric Laurent polynomials of degree <= n\leq n, as in Discussion 2.1(3). It is now time to start approximating functions by trigonometric Laurent polynomials.
Let us therefore assume we start with some object F inTF \in \mathscr{T}, and find a good approximation of it by the object E in bar((:G:))_(m)^([-m,m])E \in \overline{\langle G\rangle}_{m}^{[-m, m]}, meaning that we find a morphism E rarr FE \rightarrow F such that, in the triangle E rarr F rarr DE \rightarrow F \rightarrow D, the object DD belongs to T <= -M\mathscr{T} \leq-M for some suitably large MM.
Now we can try to iterate, and find a good approximation for DD. Thus we can look for a morphism E^('')rarr DE^{\prime \prime} \rightarrow D, with E^('')in bar((:G:)_(n))_(n,n])E^{\prime \prime} \in \overline{\langle G\rangle_{n}} \underset{n, n]}{ }, and such that in the triangle E^('')rarr D rarrD^(')E^{\prime \prime} \rightarrow D \rightarrow D^{\prime} the
object D^(')D^{\prime} belongs to T <= -N\mathscr{T} \leq-N, with N > MN>M even more enormous than MM. Can we combine these to improve our initial approximation of FF ?
To do this, let us build up the octahedron on the composable morphisms F rarrF \rightarrowD rarrD^(')D \rightarrow D^{\prime}. We end up with a diagram where the rows and columns are triangles
and in particular the triangle E^(')rarr F rarrD^(')E^{\prime} \rightarrow F \rightarrow D^{\prime} gives that E^(')E^{\prime} is an even better approximation of FF than EE was. We are therefore interested in knowing if the triangle E rarrE^(')rarrE^('')E \rightarrow E^{\prime} \rightarrow E^{\prime \prime}, coupled with the fact that E in bar((:G:)_(m))_(m,m])E \in \overline{\langle G\rangle_{m}} \underset{m, m]}{ } and E^('')in bar((:G:)_(n))^([-n,n])E^{\prime \prime} \in \overline{\langle G\rangle_{n}}{ }^{[-n, n]}, gives any information about where E^(')E^{\prime} might lie with respect to the construction of Black Box 2.2(3). Hence it is useful to know the following.
Facts 2.6. Let T\mathscr{T} be a triangulated category with coproducts. The construction of Black Box 2.2(3) satisfies
(2) Given an exact triangle E rarrE^(')rarrE^('')E \rightarrow E^{\prime} \rightarrow E^{\prime \prime}, with E in bar((:G:))_(m)^([-m,m])E \in \overline{\langle G\rangle}_{m}^{[-m, m]} and E^('')in bar((:G:))_(n)^([-n,n])E^{\prime \prime} \in \overline{\langle G\rangle}_{n}^{[-n, n]}, it follows that E^(')in bar((:G:))_(m+n)^([-m-n,m+n])E^{\prime} \in \overline{\langle G\rangle}_{m+n}^{[-m-n, m+n]}.
Combining Remark 2.5 with Facts 2.6 allows us to improve approximations through iteration. Hence part (2) of the definition below becomes natural, it iterates to provide arbitrarily good approximations.
Definition 2.7. Let T\mathscr{T} be a triangulated category with coproducts. It is approximable if there exist a t-structure (T <= 0,T >= 0)(\mathscr{T} \leq 0, \mathscr{T} \geq 0), a compact generator G inTG \in \mathscr{T}, and an integer n > 0n>0 such that
(1) GG belongs to T <= n\mathscr{T} \leq n and Hom(G,T <= -n)=0\operatorname{Hom}(G, \mathscr{T} \leq-n)=0;
(2) Every object X inT <= 0X \in \mathscr{T} \leq 0 admits an exact triangle E rarr X rarr DE \rightarrow X \rightarrow D with E inE \inbar((:G:)_(n)[-n,n])\overline{\langle G\rangle_{n}[-n, n]} and with D inT <= -1D \in \mathscr{T} \leq-1.
Remark 2.8. While part (2) of Definition 2.7 comes motivated by the analogy with Fourier analysis, part (1) of the definition seems random. It requires the tt-structure, which is our replacement for the L^(p)L^{p}-norm, to be compatible with the compact generator, which is the analog of g(x)=e^(2pi ix)g(x)=e^{2 \pi i x}. As the reader will see in Proposition 5.5, this has the effect of uniquely specifying the tt-structure (up to equivalence). So maybe a better parallel would be to fix our norm to be a particularly nice one, for example, the L^(2)L^{2}-norm on M(S^(1))M\left(\mathbb{S}^{1}\right).
Let me repeat myself: as with all new mathematics, Definition 2.7 should be viewed as provisional. In the remainder of this survey, we will discuss the applications as they now
stand, to highlight the power of the methods. But I would not be surprised in the slightest if future applications turn out to require modifications, and/or generalizations, of the definitions and of the theorems that have worked so far.
3. EXAMPLES OF APPROXIMABLE TRIANGULATED CATEGORIES
In Section 1 we gave the definition of approximable triangulated categories. The definition combines old, classical ingredients (t-structures and compact generators) with a new
theory is nonempty: we need to produce examples, categories people care about which satisfy the definition of approximability. The current section is devoted to the known examples of approximable triangulated categories. We repeat what we have said before: the subject is in its infancy, there could well be many more examples out there.
Example 3.1. Let T\mathscr{T} be a triangulated category with coproducts. If G inTG \in \mathscr{T} is a compact generator such that Hom(G,G[i])=0\operatorname{Hom}(G, G[i])=0 for all i > 0i>0, then the category T\mathscr{T} is approximable.
This example turns out to be easy, the reader is referred to [29, EXAMPLE 3.3] for the (short) proof. Special cases include
(1) T=D(R-Mod)\mathscr{T}=\mathbf{D}(R-\mathrm{Mod}), where RR is a dga with H^(i)(R)=0H^{i}(R)=0 for i > 0i>0;
(2) The homotopy category of spectra.
Example 3.2. If XX is a quasicompact, separated scheme, then the category D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X) is approximable. We remind the reader of the traditional notation being used here: the category D(X)\mathbf{D}(X) is the unbounded derived category of complexes of sheaves of O_(X)\mathscr{O}_{X}-modules, and the full subcategory D_(qc)(X)subD(X)\mathbf{D}_{\mathbf{q c}}(X) \subset \mathbf{D}(X) has for objects the complexes with quasicoherent cohomology.
The proof of the approximability of D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X) is not trivial. The category has a standard tt-structure, that part is easy. The existence of a compact generator GG needs proof, it may be found in Bondal and Van den Bergh [6, THEOREM 3.1.1(II)]. Their proof is not constructive, it is only an existence proof, but it does give enough information to deduce that part (1) of Definition 2.7 is satisfied by every compact generator (indeed, it is satisfied by every compact object). See [6, THEOREM 3.1.1(I)]. But it is a challenge to show that we may choose a compact generator GG and an integer n > 0n>0 in such a way that Definition 2.7(2) is satisfied.
If we further assume that XX is of finite type over a noetherian ring RR, then the (relatively intricate) proof of the approximability of D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X) occupies [33, SECTIONS 4 AND 5]. The little trick, that extends the result to all quasicompact and separated XX, was not observed until later: it appears in [29, LEMMA 3.5].
Example 3.3. It is a theorem that, under mild hypotheses, the recollement of any two approximable triangulated categories is approximable. To state the "mild hypotheses" precisely: suppose we are given a recollement of triangulated categories
with R\mathscr{R} and T\mathscr{T} approximable. Assume further that the category S\mathscr{S} is compactly generated, and any compact object H inSH \in \mathscr{S} has the property that Hom(H,H[i])=0\operatorname{Hom}(H, H[i])=0 for i≫0i \gg 0. Then the category S\mathscr{S} is also approximable.
The reader can find the proof in [7, THEOREM 4.1], it is the main result in the paper. The bulk of the article is devoted to developing the machinery necessary to prove the theoremhence it is worth noting that this machinery has since demonstrated usefulness in other contexts, see the subsequent articles [27,28].
There is a beautiful theory of noncommutative schemes, and a rich literature studying them. And many of the interesting examples of such schemes are obtained as recollements of ordinary schemes, or of admissible pieces of them. Thus the theorem that recollements of approximable triangulated categories are approximable gives a wealth of new examples of approximable triangulated categories.
Since this ICM is being held in St. Petersburg, it would be remiss not to mention that the theory of noncommutative algebraic geometry, in the sense of the previous paragraph, is a subject to which Russian mathematicians have contributed a vast amount. The seminal work of Bondal, Kontsevich, Kuznetsov, Lunts, and Orlov immediately springs to mind. For a beautiful introduction to the field, the reader might wish to look at the early sections of Orlov [34]. The later sections prove an amazing new theorem, but the early ones give a lovely survey of the background. In fact, the theory sketched in this survey was born when I was trying to read and understand Orlov's beautiful article.
4. APPLICATIONS: STRONG GENERATION
We begin by reminding the reader of a classical definition, going back to Bondal and Van den Bergh [6].
Definition 4.1. Let T\mathscr{T} be triangulated category. An object G inTG \in \mathscr{T} is called a strong generator if there exists an integer â„“ > 0\ell>0 with T=(:G:)_(â„“)\mathscr{T}=\langle G\rangle_{\ell}, where the notation is as in Black Box 2.2(1). The category T\mathscr{T} is called regular or strongly generated if it contains a strong generator.
The first application of approximability is the proof of the following two theorems.
Theorem 4.2. Let XX be a quasicompact, separated scheme. The derived category of perfect complexes on XX, denoted here by D^("perf ")(X)\mathbf{D}^{\text {perf }}(X), is regular if and only if XX has a cover by open subsets Spec(R_(i))sub X\operatorname{Spec}\left(R_{i}\right) \subset X, with each R_(i)R_{i} of finite global dimension.
Remark 4.3. If XX is noetherian and separated, then Theorem 4.2 specializes to saying that D^("perf ")(X)\mathbf{D}^{\text {perf }}(X) is regular if and only if XX is regular and finite-dimensional. Hence the terminology.
Theorem 4.4. Let XX be a noetherian, separated, finite-dimensional, quasiexcellent scheme. Then the category D^(b)(Coh(X))\mathbf{D}^{b}(\operatorname{Coh}(X)), the bounded derived category of coherent sheaves on XX, is always regular.
Remark 4.5. The reader is referred to [33] and to Aoki [4] for the proofs of Theorems 4.2 and 4.4. More precisely, for Theorem 4.2 see [33, THEOREM 0.5]. About Theorem 4.4: if we add the assumption that every closed subvariety of XX admits a regular alteration then the result may be found in [33, THEOREM 0.15], but Aoki [4] found a lovely argument that allowed him to extend the statement to all quasiexcellent XX.
There is a rich literature on strong generation, with beautiful papers by many authors. In the introduction to [33], as well as in [26] and [31, sEction 7], the reader can find an extensive discussion of (some of) this fascinating work and of the way Theorems 4.2 and 4.4 compare to the older literature. For a survey taking an entirely different tack, see Minami [22], which places in historical perspective a couple of the key steps in the proofs that [33] gives for Theorems 4.2 and 4.4.
Since all of this is now well documented in the published literature, let us focus the remainder of the current survey on the other applications of approximability. Those are all still in preprint form, see [27-29], although there are (published) surveys in [31, SECTIONs 8 AND 9] and in [30]. Those surveys are fuller and more complete than the sketchy one we are about to embark on. As we present the material, we will feel free to refer the reader to the more extensive surveys whenever we deem it appropriate.
5. THE FREEDOM IN THE CHOICE OF COMPACT GENERATOR AND T-STRUCTURE
Definition 2.7 tells us that a triangulated category T\mathscr{T} with coproducts is approximable if there exist, in T\mathscr{T}, a compact generator GG and a t-structure (T <= 0,T >= 0)(\mathscr{T} \leq 0, \mathscr{T} \geq 0) satisfying some properties. The time has come to explore just how free we are in the choice of the compact generator and of the tt-structure. To address this question we begin by formulating:
Definition 5.1. Let T\mathscr{T} be a triangulated category. Then two t-structures (T_(1)^( <= 0),T_(1)^( >= 0))\left(\mathscr{T}_{1}^{\leq 0}, \mathscr{T}_{1}^{\geq 0}\right) and (T_(2)^( <= 0),T_(2)^( >= 0))\left(\mathscr{T}_{2}^{\leq 0}, \mathscr{T}_{2}^{\geq 0}\right) are declared equivalent if there exists an integer n > 0n>0 such that
Discussion 5.2. Let T\mathscr{T} be a triangulated category with coproducts. If G inTG \in \mathscr{T} is a compact object and bar((:G:))^((-oo,0])\overline{\langle G\rangle}{ }^{(-\infty, 0]} is as in Black Box 2.2(2), then Alonso, JeremÃas, and Souto [1, Ñ‚HEoREM A.1], building on the work of Keller and Vossieck [16], teaches us that there is a unique t-structure (T^( <= 0),T^( >= 0))\left(\mathscr{T}^{\leq 0}, \mathscr{T}^{\geq 0}\right) with T <= 0= bar((:G:))(-oo,n]\mathscr{T} \leq 0=\overline{\langle G\rangle}(-\infty, n]. We will call this the tt-structure generated by GG, and denote it (T_(G)^( <= 0),T_(G)^( >= 0))\left(\mathscr{T}_{G}^{\leq 0}, \mathscr{T}_{G}^{\geq 0}\right).
In Black Box 2.2(2) we asked the reader to accept, as a black box, the construction passing from an object G inTG \in \mathscr{T} to the subcategory bar((:G:))^((-oo,0])\overline{\langle G\rangle}{ }^{(-\infty, 0]}. If GG is compact, then [1[1, THEoREM A.1] allows us to express this as T_(G)^( <= 0)\mathscr{T}_{G}^{\leq 0} for a unique tt-structure. We ask the reader to accept on faith that:
Lemma 5.3. If GG and HH are two compact generators for the triangulated category T\mathscr{T}, then the two tt-structures (T_(G)^( <= 0),T_(G)^( >= 0))\left(\mathscr{T}_{G}^{\leq 0}, \mathscr{T}_{G}^{\geq 0}\right) and (T_(H)^( <= 0),T_(H)^( >= 0))\left(\mathscr{T}_{H}^{\leq 0}, \mathscr{T}_{H}^{\geq 0}\right) are equivalent as in Definition 5.2.
As it happens, the proof of Lemma 5.3 is easy, the interested reader can find it in [29, Remark 0.15]. And Lemma 5.3 leads us to:
Definition 5.4. Let T\mathscr{T} be a triangulated category in which there exists a compact generator. We define the preferred equivalence class of tt-structures as follows: a tt-structure belongs to the preferred equivalence class if it is equivalent to (T_(G)^( <= 0),T_(G)^( >= 0))\left(\mathscr{T}_{G}^{\leq 0}, \mathscr{T}_{G}^{\geq 0}\right) for some compact generator G inTG \in \mathscr{T}, and by Lemma 5.3 it is equivalent to (T_(H)^( <= 0),T_(H)^( >= 0))\left(\mathscr{T}_{H}^{\leq 0}, \mathscr{T}_{H}^{\geq 0}\right) for every compact generator HH.
The following is also not too hard, and may be found in [29, PROPOSITIONS 2.4 AND 2.6].
Proposition 5.5. Let T\mathscr{T} be an approximable triangulated category. Then for any tt-structure (T <= 0,T^( >= 0))\left(\mathscr{T} \leq 0, \mathscr{T}^{\geq 0}\right) in the preferred equivalence class, and for any compact generator H inTH \in \mathscr{T}, there exists an integer n > 0n>0 (which may depend on HH and on the tt-structure), satisfying
(1) HH belongs to T <= n\mathscr{T} \leq n and Hom(H,T <= -n)=0\operatorname{Hom}(H, \mathscr{T} \leq-n)=0;
(2) Every object X inT^( <= 0)X \in \mathscr{T}^{\leq 0} admits an exact triangle E rarr X rarr DE \rightarrow X \rightarrow D with E inE \inbar((:H:))_(n)^([-n,n])\overline{\langle H\rangle}_{n}^{[-n, n]} and with D inT <= -1D \in \mathscr{T} \leq-1.
Moreover, if HH is a compact generator, (T <= 0,T >= 0)(\mathscr{T} \leq 0, \mathscr{T} \geq 0) is a tt-structure, and there exists an integer n > 0n>0 satisfying (1) and (2) above, then the tt-structure (T <= 0,T >= 0)(\mathscr{T} \leq 0, \mathscr{T} \geq 0) must belong to the preferred equivalence class.
Remark 5.6. Strangely enough, the value of Proposition 5.5 can be that it allows us to find an explicit t-structure in the preferred equivalence class.
Consider the case where XX is a quasicompact, separated scheme. By Bondal and Van den Bergh [6, THEOREM 3.1.1(II)], we know that the category D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X) has a compact generator, but in Example 3.2 we mentioned that the existence proof is not overly constructive, it does not give us a handle on any explicit compact generator. Let GG be some compact generator. From Alonso, JeremÃas, and Souto [1, THEOREM A.1], we know that the subcategory bar((:G:))^((-oo,0])\overline{\langle G\rangle}^{(-\infty, 0]} of Black Box 2.2(2) is equal to T_(G)^( <= 0)\mathscr{T}_{G}^{\leq 0} for a unique t-structure (T_(G)^( <= 0),T_(G)^( >= 0))\left(\mathscr{T}_{G}^{\leq 0}, \mathscr{T}_{G}^{\geq 0}\right) in the preferred equivalence class. But this does not leave us a whole lot wiser-the compact generator GG is not explicit, hence neither is the t\mathrm{t}-structure.
However, the combination of [33, THEOREM 5.8] and [29, LEMMA 3.5] tells us that the category D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X) is approximable, and it so happens that the t\mathrm{t}-structure used in the proof, that is, the t\mathrm{t}-structure for which a compact generator HH and an integer n > 0n>0 satisfying (1) and (2) of Proposition 5.5 are shown to exist, happens to be the standard t-structure. From Proposition 5.5, we now deduce that the standard tt-structure is in the preferred equivalence class.
6. STRUCTURE THEOREMS IN APPROXIMABLE TRIANGULATED CATEGORIES
An approximable triangulated category T\mathscr{T} must have a compact generator GG, and Definition 5.4 constructed for us a preferred equivalence class of tt-structures-namely all
those equivalent to (T_(G)^( <= 0),T_(G)^( >= 0))\left(\mathscr{T}_{G}^{\leq 0}, \mathscr{T}_{G}^{\geq 0}\right). Recall that, for any t-structure (T <= 0,T^( >= 0))\left(\mathscr{T} \leq 0, \mathscr{T}^{\geq 0}\right), it is customary to define
It is an easy exercise to show, directly from Definition 5.1, that equivalent t-structures give rise to identical T^(-),T^(+)\mathscr{T}^{-}, \mathscr{T}^{+}, and T^(b)\mathscr{T}^{b}. Therefore triangulated categories with a single compact generator, and in particular approximable triangulated categories, have preferred subcategories T^(-),T^(+)\mathscr{T}^{-}, \mathscr{T}^{+}, and T^(b)\mathscr{T}^{b}, which are intrinsic-they are simply those corresponding to any t\mathrm{t}-structure in the preferred equivalence class. In the remainder of this survey, we will assume that T^(-),T^(+)\mathscr{T}^{-}, \mathscr{T}^{+}, and T^(b)\mathscr{T}^{b} always stand for the preferred ones.
In the heuristics of Discussion 2.1(2), we told the reader that a t-structure (T <= 0,T^( >= 0))\left(\mathscr{T} \leq 0, \mathscr{T}^{\geq 0}\right) is to be viewed as a metric on T\mathscr{T}. In Definition 6.1 below, the heuristic is that we construct a full subcategory T_(c)^(-)\mathscr{T}_{c}^{-}to be the closure of T^(c)\mathscr{T}^{c} with respect to any of the (equivalent) metrics that come from t-structures in the preferred equivalence class.
Definition 6.1. Let T\mathscr{T} be an approximable triangulated category. The full subcategory T_(c)^(-)\mathscr{T}_{c}^{-} is given by
Ob(T_(c)^(-))={F inT|[" For every integer "n > 0" and for every t-structure "],[(T <= 0","T >= 0)" in the preferred equivalence class, "],[" there exists an exact triangle "E rarr F rarr D" in "T],[" with "E inT^(c)" and "D inT <= -n]}\operatorname{Ob}\left(\mathscr{T}_{c}^{-}\right)=\left\{F \in \mathscr{T} \left\lvert\, \begin{array}{c}
\text { For every integer } n>0 \text { and for every t-structure } \\
(\mathscr{T} \leq 0, \mathscr{T} \geq 0) \text { in the preferred equivalence class, } \\
\text { there exists an exact triangle } E \rightarrow F \rightarrow D \text { in } \mathscr{T} \\
\text { with } E \in \mathscr{T}^{c} \text { and } D \in \mathscr{T} \leq-n
\end{array}\right.\right\}
The full subcategory T_(c)^(b)\mathscr{T}_{c}^{b} is defined to be T_(c)^(b)=T_(c)^(-)nnT^(b)\mathscr{T}_{c}^{b}=\mathscr{T}_{c}^{-} \cap \mathscr{T}^{b}.
Remark 6.2. Let T\mathscr{T} be an approximable triangulated category. Aside from the classical, full subcategory T^(c)\mathscr{T}^{c} of compact objects, which we encountered back in Definition 1.1, we have in this section concocted five more intrinsic, full subcategories of T\mathscr{T} : they are T^(-)\mathscr{T}^{-}, T^(+),T^(b),T_(c)^(-)\mathscr{T}^{+}, \mathscr{T}^{b}, \mathscr{T}_{c}^{-}, and T_(c)^(b)\mathscr{T}_{c}^{b}. It can be proved that all six subcategories, that is, the old T^(c)\mathscr{T}^{c} and the five new ones, are thick subcategories of T\mathscr{T}. In particular, each of them is a triangulated category.
Example 6.3. It becomes interesting to figure out what all these categories come down to in examples.
Let XX be a quasicompact, separated scheme. From Example 3.2, we know that the category T=D_(qc)(X)\mathscr{T}=\mathbf{D}_{\mathbf{q c}}(X) is approximable, and in Remark 5.6 we noted that the standard t\mathrm{t}-structure is in the preferred equivalence class. This can be used to show that, for T=D_(qc)(X)\mathscr{T}=\mathbf{D}_{\mathbf{q c}}(X), we have
where the last two equalities assume that the scheme XX is noetherian, and all six categories on the right of the equalities have their traditional meanings.
The reader can find an extensive discussion of the claims above in [31], more precisely in the paragraphs between [31, PROPOSITION 8.10] and [31, THEOREM 8.16]. That discussion
goes beyond the scope of the current survey, it analyzes the categories T_(c)^(b)subT_(c)^(-)\mathscr{T}_{c}^{b} \subset \mathscr{T}_{c}^{-}in the generality of non-noetherian schemes, where they still have a classical description-of course, not involving the category of coherent sheaves. After all coherent sheaves do not behave well for non-noetherian schemes.
Remark 6.4. In this survey we spent some effort introducing the notion of approximable triangulated categories. In Example 3.2 we told the reader that it is a theorem (and not a trivial one) that, as long as a scheme XX is quasicompact and separated, the derived category D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X) is approximable. In this section we showed that every approximable triangulated category comes with canonically defined, intrinsic subcategories T^(-),T^(+),T^(b),T^(c),T_(c)^(-)\mathscr{T}^{-}, \mathscr{T}^{+}, \mathscr{T}^{b}, \mathscr{T}^{c}, \mathscr{T}_{c}^{-}, and T_(c)^(b)\mathscr{T}_{c}^{b}, and in Example 6.3 we informed the reader that, in the special case where T=D_(qc)(X)\mathscr{T}=\mathbf{D}_{\mathbf{q c}}(X), these turn out to be D_(qc)^(-)(X),D_(qc)^(+)(X),D_(qc)^(b)(X),D^("perf ")(X),D_("coh ")^(-)(X)\mathbf{D}_{\mathbf{q c}}^{-}(X), \mathbf{D}_{\mathbf{q c}}^{+}(X), \mathbf{D}_{\mathbf{q c}}^{b}(X), \mathbf{D}^{\text {perf }}(X), \mathbf{D}_{\text {coh }}^{-}(X), and D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X), respectively.
Big deal. This teaches us that the traditional subcategories D_(qc)^(-)(X),D_(qc)^(+)(X),D_(qc)^(b)(X)\mathbf{D}_{\mathbf{q c}}^{-}(X), \mathbf{D}_{\mathbf{q c}}^{+}(X), \mathbf{D}_{\mathbf{q c}}^{b}(X), D^("perf ")(X),D_("coh ")^(-)(X)\mathbf{D}^{\text {perf }}(X), \mathbf{D}_{\text {coh }}^{-}(X), and D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X) of the category D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X) all have intrinsic descriptions. This might pass as a curiosity, unless we can actually use it to prove something we care about that we did not use to know.
Discussion 6.5. To motivate the next theorem, it might help to think of the parallel with functional analysis.
Let M(R)M(\mathbb{R}) be the vector space of Lebesgue-measurable, real-valued functions on R\mathbb{R}. Given any two functions f,g in M(R)f, g \in M(\mathbb{R}), we can pair them by integrating the product, that is, we form the pairing
(:f,g:)=int fgd mu\langle f, g\rangle=\int f g d \mu
where mu\mu is Lebesgue measure. This gives us a map
where the integral is declared to be infinite if it does not converge.
We can restrict this pairing to subspaces of M(R)M(\mathbb{R}). For example, if f inL^(p)(R)f \in L^{p}(\mathbb{R}) and g inL^(q)(R)g \in L^{q}(\mathbb{R}) with (1)/(p)+(1)/(q)=1\frac{1}{p}+\frac{1}{q}=1 then the integral converges, that is, (:f,g:)inR\langle f, g\rangle \in \mathbb{R}, and we deduce a map
which turns out to be an isometry of Banach spaces.
The category-theoretic version is that on any category T\mathscr{T} there is the pairing sending two objects A,B inTA, B \in \mathscr{T} to Hom(A,B)\operatorname{Hom}(A, B). Of course, this pairing is not symmetric, we have to keep track of the position of AA and of BB in Hom(A,B)\operatorname{Hom}(A, B). If RR is a commutative ring and T\mathscr{T} happens to be an RR-linear category, then Hom(A,B)\operatorname{Hom}(A, B) is an RR-module and the pairing delivers a map
where the op keeps track of the variable in the first position. And now we are free to restrict to subcategories of T\mathscr{T}.
If T\mathscr{T} happens to be approximable and RR-linear, we have just learned that it comes with six intrinsic subcategories T^(-),T^(+),T^(b),T^(c),T_(c)^(-)\mathscr{T}^{-}, \mathscr{T}^{+}, \mathscr{T}^{b}, \mathscr{T}^{c}, \mathscr{T}_{c}^{-}, and T_(c)^(b)\mathscr{T}_{c}^{b}. We are free to restrict the Hom pairing to any couple of them. This gives us 36 possible pairings, and each of those yields two maps from a subcategory to the dual of another. There are 72 cases we could study, and the theorem below tells us something useful about four of those.
Theorem 6.6. Let RR be a noetherian ring, and let T\mathscr{T} be an RR-linear, approximable triangulated category. Suppose there exists in T\mathscr{T} a compact generator GG so that Hom(G,G[n])\operatorname{Hom}(G, G[n]) is a finite RR-module for all n inZn \in \mathbb{Z}. Consider the two functors
defined by the formulas Y(B)=Hom(-,B)\mathscr{Y}(B)=\operatorname{Hom}(-, B) and widetilde(Y)(A)=Hom(A,-)\widetilde{\mathscr{Y}}(A)=\operatorname{Hom}(A,-), as in Discussion 6.5 . Now consider the following composites:
(1) The functor Y\mathscr{Y} is full, and the essential image consists of the locally finite homological functors (see Explanation 6.7 for the definition of locally finite functors). The composite Y\mathscr{Y} â—‹ ii is fully faithful, and the essential image consists of the finite homological functors (again, see Explanation 6.7 for the definition).
(2) With the notation as in Black Box 2.2(1), assume ^(1){ }^{1} that T= bar((:H:))\mathscr{T}=\overline{\langle H\rangle} for some integer n > 0n>0 and some object H inT_(c)^(b)H \in \mathscr{T}_{c}^{b}. Then the functor widetilde(Y)\widetilde{\mathscr{Y}} is full, and the essential image consists of the locally finite homological functors. The composite widetilde(Y)\widetilde{\mathscr{Y}} ○ г̃ is fully faithful, and the essential image consists of the finite homological functors.
Explanation 6.7. In the statement of Theorem 6.6, the locally finite functors, either of the form H:[T^(c)]^("op ")rarr RH:\left[\mathscr{T}^{c}\right]^{\text {op }} \rightarrow R-Mod or of the form H:T_(c)^(b)rarr RH: \mathscr{T}_{c}^{b} \rightarrow R-Mod, are the functors such that
(1) H(A[i])H(A[i]) is a finite RR-module for every i inZi \in \mathbb{Z} and every AA in either T^(c)\mathscr{T}^{c} or T_(c)^(b)\mathscr{T}_{c}^{b}
